Posts Tagged ‘quantum mechanics’

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

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Consistent Quantum Theory

September 4, 2007

Quantum mechanics is a notoriously confusing and difficult subject, which is terribly unfortunate given its status as a fundamental theory of how the world works. When I was an undergraduate and graduate student of physics in the 1980’s, I was always unsatisfied with the way the subject was presented. We basically learned how to calculate solutions to certain problems, and there was an understanding that you were better off just avoiding the philosophical issues or the paradoxes surrounding the subject. After all, everybody knew that all the challenges raised by physicists as eminent as Einstein had ultimately been resolved experimentally in favor of the “Copenhagen interpretation.”

Still, there was a lot of weird stuff in the Copenhagen interpretation like the “collapse of the wave-function” caused by a measurement, or apparently non-local effects. What I wished existed was a clearly-defined and sensible set of “rules of the game” for using quantum mechanics. The need for such a set of rules has only increased with time with the advent of the field of quantum computing.

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In his book, “Consistent Quantum Theory,” (the first 12 out of 27 chapters are available online) noted mathematical physicist Robert Griffiths provides the textbook I wished I had as a student. With contributions from Murray Gell-Mann, Roland Omnes, and James Hartle, Griffiths originated the “consistent history” approach to quantum mechanics which is explained in this book, .

The best summary of the approach can be obtained from the comparison Griffiths makes between quantum mechanics and the previous classical mechanics. Quantum mechanics differs from classical mechanics in the following ways:

1. Physical objects never posses a completely precise position or momentum.
2. The fundamental dynamical laws of physics are stochastic and not deterministic, so from the present state of the world one cannot infer a unique future (or past) course of events.
3. The principle of unicity does not hold: there is not a unique exhaustive description of a physical system or a physical process. Instead reality is such that it can be described in various alternative, incompatible ways, using descriptions which cannot be combined or compared.

It is the 3rd point which is the heart of the approach. In quantum mechanics, you are permitted to describe systems using one “consistent framework” or another, but you may not mix incompatible frameworks (basically frameworks using operators that don’t commute) together.

Griffiths uses toy models to illustrate the approach throughout the book, and provides resolutions for a large number of quantum paradoxes. He stresses that measurement plays no fundamental role in quantum mechanics, that wave function collapse is not a physical process, that quantum mechanics is a strictly local theory, and that reality exists independent of any observers. All of these points might seem philosphically natural and unremarkable, except for the fact that they contradict the standard Copenhagen interpretation.

This is a book that will take some commitment from the reader to understand, but it is the best explanation of quantum mechanics that I know of.

Algorithms for Physics

August 14, 2007


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Much of my own work is at the intersection of statistical mechanics and algorithms, in particular understanding and developing new algorithms using ideas originating in statistical mechanics. Werner Krauth also works at the intersection of the two fields, but coming from a very different angle: he is a leading expert on the development and application of algorithms to compute and understand the properties of physical systems.

In his recently published book, “Statistical Mechanics: Algorithms and Computations,” targeted at advanced undergraduates or graduate students, he covers a very wide range of interesting algorithms. To give you an idea of the coverage, I’ll just list the chapters: “Monte Carlo methods,” “Hard disks and spheres,” “Density matrices and path integrals,” “Bosons,” “Order and disorder in spin systems, “Entropic forces,”and “Dynamic Monte Carlo methods.”

Krauth’s presentation is leavened by his humor, and he often uses the results obtained using his algorithms to make surprising points about physics that would otherwise be hard to convey.

I am often asked by computer science or electrical engineering scientists and researchers for good introductions to physics, and particularly statistical mechanics, and I’m now happy to be able to recommend this book.

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Specifying physics explicitly in terms of algorithms, as Krauth does, gives a very concrete basis for understanding concepts that can otherwise seem terribly abstract. Gerald Sussman and Jack Wisdom make this point in the preface of their already classic book (which is available online) “Structure and Interpretation of Classical Mechanics”:

“Computational algorithms are used to communicate precisely some of the methods used in the analysis of dynamical phenomena. Expressing the methods of variational mechanics in a computer language forces them to be unambiguous and computationally effective. Computation requires us to be precise about the representation of mechanical and geometric notions as computational objects and permits us to represent explicitly the algorithms for manipulating these objects. Also, once formalized as a procedure, a mathematical idea becomes a tool that can be used directly to compute results.”

But while Sussman and Wisdom’s book focuses in great detail on classical mechanics, Krauth’s book covers more broadly subjects in classical mechanics, statistical mechanics, quantum mechanics, and even quantum statistical mechanics. Another difference is that Sussman and Wisdom specify their algorithms in executable Scheme code, while Krauth uses pseudo-code. Of course, both choices have their advantages, just as both of these books are worth your time.