Random Sampling and Feedback Does the Trick
A new idea for sampling Feynman diagrams came to Prokof’ev and Svistunov in 1998 — but prevailing thinking said it too was impossible for macroscopic systems. Overcoming their own doubts, they began their project for strongly-interacting fermions in 2008.
Both the boson polymer approach and Feynman diagrams give physicists a way to calculate the statistics. With the polymer approach, the sum is over all possible paths. But with Feynman diagrams, the sum is only over virtual interaction events.
Nonetheless, step-by-step calculation of each Feynman diagram in many-body systems is impossible due to the sheer number required. So the UMass team samples Feynman diagrams at random, and feeds the results back into its computer simulation. In this way, the process slowly converges. They call it the Bold Diagrammatic Monte Carlo (BDMC) method.
Svistunov explained to Decoded Science that it is like polling voters for an upcoming election (sampling), telling them the results of the poll (feedback), and then repolling the same voters. Their knowledge of the original poll results influences their new poll choices.
More formally, according to the Amherst’s press release of March 18, 2012:
“We poll a series of integrals, and the result is fed back to the series to keep improving our knowledge . . . ,” says Kris Van Houcke, who developed the BDMC code over the past three years.
“We repeat this with several hundred processors over several days until the solution converges,” Prokof’ev and Svistunov add. Once completed, “you know all the basic thermodynamic properties of the system. This has never been done before.”
Experiments conducted by Martin Zwierlein and colleagues at MIT confirm the sampling/feedback approach. Per the press release:
“Our answers and the experimental results perfectly agree,” Van Houcke said. “Our new method makes accurate predictions.”
“The accompanying experiment is a breakthrough on its own,” Svistunov added, “because achieving a few percent accuracy has long been a dream in the field of ultra-cold atoms. We needed this confirmation from Mother Nature.”
The Prokof’ev/Svistunov approach has potential applications in “currently intractable problems in high temperature superconductivity” as well as “many-body problems in high-energy physics, condensed matter and ultra-cold atoms.
The approach is universal,” Svistunov told Decoded Science. “It applies to all fermionic systems, including neutron stars. It also applies to exotic bosonic and spin systems when the mapping into polymers does not work (where one can represent them in terms of fermions).” And it could lead to a deeper understand in areas of the quantum world no one has yet thought of.
Acknowledgement: The author wishes to thank Dr. Svistunov for his review and recommendations for this article.
Lathrop, J. UMass Amherst Theoretical Physicists Find a Way to Simulate Strongly Correlated Fermions. Accessed March 23, 2012.
Feynman, R. P., QED, The Strange Theory of Light and Matter (1985). Princeton: Princeton University Press.
Van Houcke, K. et al. Feynman diagrams versus Fermi-gas Feynman emulator. (2012). Nature Physics. Accessed March 23, 2012.
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