Andy Schofield - Future Conference Talk

The Future Conference: the New Scientist and Jesus College, Cambridge
28-29 August 1997
Session 4: Physics 2

New states of correlated electrons

Andy Schofield, Cavendish Laboratory, Cambridge

These are exciting times for this field of study because we believe we are now standing at the threshold of a revolution in how we view the behaviour of electrons in metals. This revolution has been created by many of the changes that are being described by others at this symposium. As we look at more complex systems, we don't see straightforward extrapolations of what we see in simple systems. Instead, we see new types of behaviour which require new concepts in order to understand them. Often called "emergent" phenomena, their character was neatly summed up by Sir Arthur Stanley Eddington when he said:

"We used to think that if we knew one, we knew two, because one and one are two. We are finding we must learn a great deal more about 'and'."

This century has seen the remarkable success of the one-electron picture of metals. This has underpinned our understanding of the metallic state, and its successes are exemplified by the semiconductor revolution. Now its foundations are being undermined by the discovery of materials which do not fit this pattern. Among the examples of these new materials are the high-temperature superconductors.

Let me give you a simple view of the ingredients which go into the one-electron picture of metals. A metal is like a gas of electrons. Each electron has two properties: a negative electric charge and its magnetic spin--it has its own internal compass which can either point North or South. The rules that the electron obeys in a metal are firstly laid down by quantum mechanics (particles should be viewed as waves and only certain wavelengths are allowed), and by the Pauli principle (the maximum number of electrons which can be assigned to a given wavelength is two and then the two electrons must have opposite spin). The final feature included in the one-electron picture is the lattice. Each electron has come from an atom which forms an ordered lattice of positive charge--positive because the atoms have given up a negative electron.

The crucial point, though, is that the repulsive interactions between electrons have been left out in the one-electron view. Each electron is negatively charged and will repel other electrons. There are of course many electrons in a metal, but in the one-electron picture, the movement of the electrons is treated as though each one is alone. The success of this picture has been one of the triumphs of physics this century. But first, a brief history of metals:

A Brief History of Metals

This is subjective and is bound to leave out a vast number of important discoveries--in this case, and among other things, magnetism. The fact that metals are different from insulators has been known since Bronze Age man first picked up a bronze rod from the fire. While you can happily hold the other end of a burning stick, holding onto a metal rod in a fire is a big mistake.

With the discovery of the electron a hundred years ago, it was soon realised that metals conducted heat because in metals electrons are free to move and carry electricity and heat. The development of quantum mechanics in the 1920s was also quickly applied to electrons in metals, and then all the ingredients of the one-electron picture were in place. Note in passing how quickly developments in particle physics were exploited in condensed matter physics: the electron was discovered in 1897 and electron theory was applied to metals in 1900; quantum mechanics was discovered in 1926 and was applied to metals in 1928. This trend continues (and can work the other way too). Twenty years on from that, the one-electron picture of metals helped develop useful technology. The transistor and semiconductor revolution is a by-product of the one-electron picture. Experiment often proceeds theory. A theoretical understanding of superconductivity (discovered in 1911) eluded some of the greatest minds--including Heisenberg and Feynman--but it was eventually understood (in 1957) as a phase transition of a one-electron metal.

The success of the one-electron picture was a big surprise but was rationalised by Landau in 1956. The theory works well because it can include some of the effects of repulsion between electrons even though it deals with single electrons. Using a modified mass for the electron is one way in which the one-electron picture can mimic the effect of the other electrons. When an electron moves, it has to push others out of the way which makes it appear heavier. So by simply increasing the electron's mass it is possible to include some of the effects of interaction. The discovery (in 1979) of metals where the electron appears to be thousands of times heavier than a lone electron is a remarkable confirmation of this idea. In recent years, we have been applying the computer to make predictions based on the one-electron picture. This is a trend which will continue.

This century has been marked by the success of the one-electron picture, but as the century closes, material scientists and chemists are producing more complicated materials which defy explanation within this picture. In the next century we will see the growing importance of the `many-electron' picture when we need to understand the full consequences of interaction between electrons.

High-temperature superconductors are the best known new materials which have properties that cannot be understood by the one-electron picture. Another phenomenon is the fractional quantum Hall effect--whose explanation lies totally outside the one-electron picture and shows an incredible richness.


A further example is provided by the "incoherent" metal UBe13 which has a complex structure with 112 atoms in each unit cell. Its strange behaviour undermines what we thought we knew about metals and also requires us to rethink our understanding of the superconducting transition. Close to absolute zero, UBe13 is a superconductor. Superconductivity is a phase transition of a one-electron metal. So just like melting ice into water, as I heat up this superconductor I should get a one-electron metal. Its resistance to electric current should increase as the temperature rises. With the superconductor UBe13 you find that as you heat it up from close to absolute zero, at first its resistance leaps to a huge value as it loses its superconducting properties. But then its resistance gradually falls again as you heat it further. This is totally contrary to how one-electron metals should behave.

Here, then, is a metal we don't understand. Looking to the future, what then is likely to come out of a new theory of metals? One of the more exciting things we might see in a more complete theory of metals is the appearance of new types of particle. It will make less sense to talk about electrons, but rather we will use other objects to describe the metallic state.
Here is one example of where strong interaction can appear to separate the electric charge and magnetism of a single electron into two distinct entities. Imagine a chain of atoms where the electrons are strongly repelled. The effect of the strong interactions leads to two new rules governing the behaviour of the electrons. The first rule is that because of the strong repulsion, rather than the two electrons that each atom could hold and still obey the Pauli principle, cramming two electrons onto one atom costs too much energy and so is forbidden. Only one electron is allowed per atom. The second rule is that the magnetic direction of neighbouring electrons should oppose each other giving an alternating magnetic arrangement.

Diagram 1

I start with one negative electron in every positive atom so the whole system is electrically neutral. Every up electron has a down electron neighbour so the whole system is also nonmagnetic. I now remove a single electron, which I could do by shining light on it. In so doing, I remove one unit of electrical charge and one unit of magnetism since both are bound up in the properties of an electron.

Diagram 2

So, while previously I couldn't move electrons around without breaking rule one, now I can move them. As electrons move, the situation evolves as follows:

Diagram 3

As electrons hop into the empty site their charge moves but the magnetic distortion (where rule two is broken) remains fixed. Eventually the place where the charge imbalance has occurred is in a different place from where the magnetic imbalance has occurred. In this system it makes more sense to talk about the magnetic and charge parts of the electron as separate entities. We call them spinons and holons. It is important to realize that the electron by itself is stable, but what has happened here is that the presence of other electrons have caused completely new types of excited states to appear which can not be mimicked by a single electron: these are new particles. The fractional quantum Hall effect provides another example of this phenomenon. There the new particles that arise carry fractions of an electrons charge.

In this century we have tamed the single electron: the semiconductor industry is a testament to how we can tailor electronic devices to control the properties of single electrons. As I hope I've illustrated already, I believe that the next century will be characterised by learning and exploiting what lies beyond one-electron physics. I've hinted at the possibility of new types of particles which may help us to picture these metals. If these could then be harnessed we might expect a new technology every bit as rich as that of the semiconductor industry. We might one day control independently the spin and charge aspect of electrons among other things. At present computers rely on the charge of the electron to transmit information and the magnetism of the electron to store information - both operating in base-two (binary). Independent control of the spin and charge of an electron could perhaps allows us to use both aspects at once and create computers working in base-four instead of binary.

Another possible technological development is that of small scale nanomagnets. These offer the possibility of coupling to the magnetism, not of the electron but the atomic nucleus - important perhaps for quantum computing.

Where will the ideas for understanding these materials come from? I've already mentioned the symbiotic relationship of high-energy particle physics and this field of condensed matter physics. Ideas such as broken symmetry and the Higgs mechanism were first understood in condensed matter systems before being applied to particle physics. Working the other way, ideas similar to the asymptotic freedom of quarks form the basis of how we understand the heavy electron metals I mentioned earlier. The interplay between these two areas of physics will continue to flourish.

One area in high-energy physics ripe for exploitation in condensed matter is that of supersymmetry and duality. While I don't have time to explain this idea in detail, striking progress has been made recently in understanding certain problems where interactions between elementary particles are very strong. An example of its success is in determining the entropy of a black hole. If the ideas of supersymmetry and duality could be applied to the interactions of the many-electron problem, this could give us tremendous insight into the physics of the new metals.

To conclude, a quiet revolution is happening in our view of the metallic state. Revolutions can be dangerous places to be and making predictions is one way to get hurt, but they can be tremendously exciting.