1. Introduction

  • The term 'strongly correlated' in many-body physics is generally applied to materials where, in some fairly vague sense, 'Coulomb interactions are important'. The methods used in modern electronic structure calculations (e.g. density functional theory) are not controlled by coupling constants originating in the interaction strength and they contain reasonably good approximate Coulomb interactions, so why is it that historically band theory has had serious problems in describing strongly-correlated materials?

  • Studies of simplified models, such as the Hubbard model, have been very fruitful in providing an understanding of the basic processes underlying the physics of strongly correlated materials.

  • Can we find a good 'one-electron' approximate method that will complement the model work by telling us something useful about real materials without inquiring too deeply into processes dependent on the rather subtle quantum fluctuations?

  • What is a strongly correlated material, really?


Hello. This talk is supposed to be a bit of background to Peter Littlewood's course on correlated electron physics. It will be largely aimed at graduate students but looking at the subject from a slightly different angle. That angle is first principles calculations, which are routinely used to calculate accurate properties of 'real' materials by solving the Schrodinger equation on a computer using well-defined approximate methods, beginning only from a knowledge of the atomic numbers of the atoms involved and from (in most cases) the crystal structure. Historically, there is a class of materials - the 'strongly correlated' materials, which are not amenable to such a treatment. Here is an extract from a book summarizing the position in the case of the parent compounds of the high-Tc superconding oxides, which are among the most well-known strongly correlated materials:

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It will be my contention that most of the statements in this extract are not true, or at the very least, misleading. However, it is an inescapable fact that (in most cases) 'one-electron band calculations predict metallic behaviour for ... even the ones that are insulators'. Why is this? As we shall see, it is in fact an artefact of the use of the approximate functionals commonly used in density functional theory, one of the most popular ways of performing band calculations. It is not because of the 'one-electron' approximation itself.

Theoretical people who study these materials are traditionally divided into camps, neither of whom ever speak to the other. On one side, in the red corner, you might say, we have the field theorists/many body theorists. In the blue corner we have the electronic structure people, people like me who spend their lives playing with CASTEP or other popular codes trying to solve the Schrodinger equation with density functional theory and to a lesser extent things like Hartree-Fock and quantum Monte Carlo and other techniques.

Now, what are the barriers to communication between these groups? Electronic structure people don't talk to many-body theory people because in general they don't understand what it is they do. Many-body theory people don't talk to us because they don't think we can tell them anything useful. Which is true up to a point. 'What good is a technique if it only works some of the time?' is a common complaint.

So now let's look a little deeper at strongly-correlated materials, and to look at the current state of the art in electronic structure calculations of these things..

OK, now we start from the premise that you can't treat strongly correlated systems with band theory. OK. But one's first intuition is that they are just quantum mechanical systems like any other that have to obey the Schrodinger equation. Obviously were forced to make sometimes drastic approximations to solve the Schrodinger equation on the computer, but these work well for silicon or diamond or other simple systems. Which one of our approximations is causing things to screw up in the high Tc cuprates etc.? If we find we can get reliable answers, we want to see if we can build on the work that's been done on model systems, and to see if we can get any insight into real materials.

Now, we have to remember what were trying to do here. - rather than trying to fit experimental data, you can think of an electronic structure calculation itself as a computer experiment. It's rather like trying to explain general relativity by saying you take a brick, you do this to it, this is what happens. But we have an electronic structure model of some real material instead of a brick.

An example of what we might like to do. Lanthanum cuprate is a rather boring antiferromagnetic insulator, except if you dope it with strontium beyond a certain level it become superconducting. What we can do in principle is to take our model of that, and we can dope it with strontium, and we can do experiments on it. We ought to be able to see holes form in the CuO2 plane, we could work out their energies of interaction, we can watch what happens to the antiferromagnetic interactions between the copper ions in the plane when the holes appear. Do we see spin polarons? Great. If we could do it. But of course, we have to recognise our limitations. What such a model is not going to do is to go through a superconducting transition, because usually we approximated out all the important maths. That of course, needs a properly formulated many body theory. But it would be very interesting if we could do all the other stuff, but we know that in general we haven't been able to.