COMPUTING SCIENCE

# Hip-Hop Physics

Electrons dance to a quantum beat in the Hubbard model of solid-state physics

# Back to the Lab Bench

The newest trick in the long struggle to master the Hubbard model is the idea of setting up experimental apparatus to mimic the model’s lattice and its population of electrons. Standing in for the electrons in these experiments are atoms of an ultracold gas—so cold, and therefore so sluggish, that the atoms can be trapped by the feeble electromagnetic field of a beam of light. Three such beams crossing at right angles create a sort of three-dimensional egg carton, with periodic wells where the cold atoms tend to accumulate. The atoms chosen for the experiment are members of the same quantum-mechanical class as electrons (known as *fermions*), and so they are subject to the same quantum rules as the electrons in a solid; in particular, no two atoms in the same spin state can occupy the same site in the optical lattice. Furthermore, two atoms of differing spin that land on the same site experience a short-range repulsion, just as electrons do. Thus the stage is set for a full simulation of the Hubbard model. The optical lattice is typically a few millimeters across, and it holds tens of thousands of atoms.

Of course it’s easy to talk about the principles of such an experiment; making it happen in the laboratory is harder—indeed, heroic. But in recent months three groups have reported successful results of such experiments. Two of those groups have found evidence of a Mott-insulator phase in the confined atoms. (The groups are led by Immanuel Bloch of Johannes Gutenberg University in Mainz, Germany, and by Robert Jördens and Niels Strohmaier of the Swiss Federal Institute of Technology in Zurich.) The third group, led by Gyu-Boong Jo of MIT, has seen signs of ferromagnetic order in a gas of cold atoms.

There is something deliciously involuted about this turn of events: an experiment that illuminates a model that explicates an experiment. One interpretation is that the assembly of lasers and cryogenic atoms and other apparatus is acting as a special-purpose quantum computer, which can efficiently solve problems that would require exponentially greater effort on a classical computer.

This audacious approach to doing science—harnessing physics to do the work of mathematics and computation—surely has great promise; and yet I have a nagging reservation. As I said at the beginning of this essay, a mathematical model is an aid to understanding, not just an engine for producing answers. According to this view, the aim of the Hubbard model is not so much to determine whether certain kinds of solids have ferromagnetic order or Mott transitions or superconducting phases. What the model offers is hope of understanding where those traits come from—how the basic ingredients of the model combine to yield emergent properties. The cold-atom experiments seem less effective as aids to understanding: Even when they yield the right answer, we may not easily see *why* it’s right. The experiments get their power from the same quantum mysteries they seek to explain.

The same complaint has often been made about ordinary computer simulations and about computer-aided proofs in mathematics. If the computer is just a black box, and we cannot follow along step by step from premise to conclusion, how can we pretend to understand the result? But these doubts about the legitimacy of computer-aided science seem to be fading, chased away by the spread of algorithmic thinking. Perhaps now we need to wait for a wider embrace of quantum thinking.

# To my readers

After the publication of this column (my 97th since 1993), I will be taking a one-year sabbatical leave from the *Computing Science* department. Although I have a long list of topics for columns that I’m still eager to learn about and write about, I also have other projects I’ve deferred too long. After a year’s respite from bimonthly deadlines, I look forward to returning to this splendid soapbox in 2011. In the meantime, this space will be filled with fresh voices and viewpoints. For those who wish to follow my own ongoing adventures, I expect to continue posting occasional essays at bit-player.org.—*Brian Hayes*

©Brian Hayes

# Bibliography

- Creutz, Michael. 2003. Direct simulations of small multi-fermion systems.
*International Journal of Modern Physics*C14:1027–1040. - Essler, Fabian H. L., Holger Frahm, Frank Göhmann, Andreas Klümper and Vladimir E. Korepin. 2005.
*The One-Dimensional Hubbard Model*. Cambridge: Cambridge University Press. - Gutzwiller, M. C. 1963. Effect of correlation on the ferromagnetism of transition metals.
*Physical Review Letters*10:159–162. - Hirsch, J. E. 1985. Two-dimensional Hubbard model: Numerical simulation study.
*Physical Review B*31:4403–4419. - Hubbard, J. 1963. Electron correlations in narrow energy bands.
*Proceedings of the Royal Society of London Series A (Mathematical and Physical Sciences)*276(1365):238–257. - Jafari, S. Akbar. 2008. Introduction to Hubbard model and exact diagonalization. http://arxiv.org/abs/0807.4878
- Jo, Gyu-Boong, Ye-Ryoung Lee, Jae-Hoon Choi, Caleb A. Christensen, Tony H. Kim, Joseph H. Thywissen, David E. Pritchard and Wolfgang Ketterle. 2009. Itinerant ferromagnetism in a Fermi gas of ultracold atoms.
*Science*325:1521–1524. - Jördens, Robert, Niels Strohmaier, Kenneth Günter, Henning Moritz and Tilman Esslinger. 2008. A Mott insulator of fermionic atoms in an optical lattice.
*Nature*455:204–208. - Lieb, Elliott H., and F. Y. Wu. 1968. Absence of Mott transition in an exact solution of the short-range one-band model in one dimension.
*Physical Review Letters*20:1445–1448. - Lieb, Elliott H., and F. Y. Wu. 2003. The one-dimensional Hubbard model: A reminiscence.
*Physica A*321:1–27. - Montorsi, Arianna (ed.). 1992.
*The Hubbard Model: A Reprint Volume*. Singapore: World Scientific. - Quintanilla, Jorge, and Chris Hooley. 2009. The strong-correlations puzzle.
*Physics World*22(6):32–37. - Rasetti, Mario (ed.). 1991.
*The Hubbard Model: Recent Results*. Singapore: World Scientific. - Schneider, U., L. Hackermüller, S. Will, Th. Best, I. Bloch, T. A. Costi, R. W. Helmes, D. Rasch and A. Rosch. 2008. Metallic and insulating phases of repulsively interacting fermions in a 3D optical lattice.
*Science*322:1520–1525. - Sorella, S., S. Baroni, R. Car and M. Parrinello. 1989. A novel technique for the simulation of interacting fermion systems.
*Europhysics Letters*8:663–668. - Tasaki, Hal. 1995, 1997. The Hubbard model: Introduction and selected rigorous results. http://arxiv.org/abs/cond-mat/9512169
- White, S. R., D. J. Scalapino, R. L. Sugar, E. Y. Loh, J. E. Gubernatis and R. T. Scalettar. 1989. Numerical study of the two-dimensional Hubbard model.
*Physical Review B*40:506–516. - Yedidia, Jonathan S. 1990. Thermodynamics of the infinite-
*U*Hubbard model.*Physical Review B*41:9397–9402. - Yedidia, Jonathan. 2007. The Hubbard model. http://nerdwisdom.com/tutorials/the-hubbard-model/

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