Posts Tagged ‘superconductivity’

The Hubbard Model: a Tutorial

September 9, 2007

Today, I will introduce a new kind of post at Nerd Wisdom, that I hope to do more of in the future: the tutorial. These posts will be a little more technical than ordinary posts, and will probably run a little longer as well. They are designed to give the intelligent and well-educated scientist an entry into a field for which he or she is not already a specialist.

When working with graduate-student interns, I find myself presenting these types of tutorials constantly. Unfortunately, this type of material is very hard to find in the literature, because it covers material that is too well-known to the specialist, while sometimes being beyond what is available in textbooks. Thus, there tends to be an artificial barrier to entry into new fields.

My first tutorial will cover the Hubbard Model. To keep this post to a reasonable length, I will just include the first part of the tutorial here.


I previously discussed high-temperature superconductivity in cuprates, and mentioned that the detailed mechanism is still controversial. However, what is widely agreed is that, as originally proposed by P.W. Anderson, a good model for these materials is the Hubbard Model (see this paper by Anderson for an entertaining and readable argument in favor of this point). And even if one doesn’t agree with that statement, the Hubbard Model is of enormous intrinsic interest, as perhaps the simplest model of interacting electrons on a lattice.

Despite its simplicity, physicists have not been able to solve for the behavior of the two or three-dimensional Hubbard model in the “thermodynamic limit” (for lattices with a very large number of sites and electrons). Coming up with a reliable approach to solving the Hubbard model has become a kind of holy grail of condensed matter theory.

Note that this is very different from the situation for the classical ferromagnetic Ising model, for which Onsager solved the two-dimensional version exactly in 1944 but where we do not have an exact solution in three dimensions. For the three-dimensional Ising model, we may not have an exact solution, but we understand extremely well the qualitative behavior, and can compute quantitative results to practically arbitrary accuracy. For the Hubbard model, we do not even have know what the qualitative behavior is for two or three dimensional lattices.

I will first describe the model in words, and then show you how to solve for the quantum statistical mechanics of the simplest non-trivial version of the model: the Hubbard model on a lattice with just two sites. I strongly believe that whenever you want to learn about a new algorithm or theory, you should start by solving, and understanding in detail, the smallest non-trivial version that you can construct.

The Full Tutorial


High-Temperature Superconductivity

September 7, 2007

In 1986, a new class of materials, called “cuprate superconductors,” was discovered by Karl Muller and Johannes Bednorz, which displayed superconductivity (the flow of electricity at zero resistance) at much higher temperatures than had ever previously been found. The discovery raised the possibility of materials that could super-conduct at ordinary room temperatures, which would clearly have great technological implications. Muller and Bednorz were awarded the Nobel Prize in Physics in 1987, which stands as the shortest period ever between a discovery and a Nobel prize.

I was a graduate student at Princeton when the news of high-temperature superconductivity broke, and my Ph.D. advisor was Philip W. Anderson, the 1977 Nobel Laureate in physics who at the time was already probably the most famous and respected condensed matter theorist in the world. (One recent study claimed to show that Anderson is the “most creative physicist in the world;” I found the method of the study highly dubious, but the conclusion is not unreasonable.) Anderson nearly immediately proposed a version of his “Resonating-Valence-Bond” theory for the superconductors, and trying to develop a complete theory has been his primary preoccupation ever since.


To give you an idea of the excitement in 1987, take a look at this article, looking back at the March 1987 meeting of the American Physical Society in New York City, characterized as the “Woodstock of Physics.”

I was at that meeting, and as one of Anderson’s students, I was definitely at the epicenter of the excitement, but I was actually rather turned off by the whole atmosphere, and I avoided working on the subject too seriously, focusing instead on neural networks and spin glass theory. (As I mentioned in this post, Anderson also made seminal contributions to those fields, and he let me work on whatever I was interested in.) Nevertheless, it was impossible to avoid being influenced, and I did eventually work on a few papers related to the theory of high-temperature superconductivity.

Sadly, the hopes for room temperature cuprate super-conductivity faded with time, and no cuprates have been discovered which super-conduct above 138 degrees Kelvin. And although our understanding of the materials has improved with time, there is still also considerable controversy about the mechanism causing the superconductivity.

See this post for more information about the Hubbard Model, which underlies the physics of the cuprates.