Simulating strongly correlated fermions opens the door to practical superconductor applications

March 18, 2012
This is Boris Svistunov. Credit: Courtesy of UMass Amherst

Combining known factors in a new way, theoretical physicists Boris Svistunov and Nikolai Prokof'ev at the University of Massachusetts Amherst, with three alumni of their group, have solved an intractable 50-year-old problem: How to simulate strongly interacting quantum systems to allow accurate predictions of their properties.

It could open the door to practical superconductor applications, as well as to solving difficult "many-body" problems in , and ultra-cold atoms.

The theoretical breakthrough by Prokof'ev and Svistunov at UMass Amherst, with their alumni Kris Van Houcke now at Ghent University, Felix Werner at Ecole Normale Supérieure Paris and Evgeny Kozik at Ecole Polytechnique, is reported in the current issue of Nature Physics. The paper also includes crucial results of an experimental validation conducted by Martin Zwierlein and colleagues at MIT.

Svistunov says, "The accompanying experiment is a breakthrough on its own 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."

Van Houcke adds, "Our answers and the experimental results perfectly agree. This is important because in physics you can always make a prediction, but unless it is controlled, with narrow error bars, you're basically just gambling. Our new method makes accurate predictions."

Physicists have long been able to numerically simulate statistical behavior of bosonic systems by mapping them onto polymers in four dimensions, as Richard Feynman proposed in the 1950s. "In a bosonic liquid one typically wants to know at what temperature the superfluid phase transition occurs," Prokof'ev explains, "and mapping onto the polymers yields an essentially exact answer."

But simulating particle behavior in strongly interacting fermionic liquids, like strongly interacting electrons in high-temperature superconducting compounds, has been devilishly elusive, he adds. "The polymer trick does not work here because of the notorious negative-sign problem, a hallmark of fermionic statistics."

Apart from mapping onto the polymers, Feynman proposed yet another solution, in terms of "diagrams" now named after him. These Feynman diagrams are graphical expressions for serial expansion of Green's functions, a mathematical tool that describes statistical properties of each unique system. Feynman diagrams were never used for making quantitatively accurate predictions for strongly interacting systems because people believed that evaluating and summing all of them was simply impossible, Svistunov points out. But the UMass Amherst team now has found a way to do this.

What they discovered is a trick—called Diagrammatic Monte Carlo—of sampling the Feynman series instead of calculating diagrams one by one. Especially powerful is the Bold Diagrammatic Monte Carlo (BDMC) scheme. This deals with a partially summed Feynman series (Dyson's development) in which the diagrams are constructed not from the bare Green's functions of non-interacting system (usually represented by thin lines), but from the genuine Green's functions of the strongly interacting system being looked for (usually represented by bold lines).

"We poll a series of integrals, and the result is fed back to the series to keep improving our knowledge of the Green's function," says Van Houcke, who developed the BDMC code over the past three years.

The BDMC protocol works a bit like sampling to predict the outcome of an election but with the difference that results of polling are being constantly fed back to the "electorate," Prokof'ev and Svistunov add. "We repeat this with several hundred processors over several days until the solution converges. That is, the Green's function doesn't change anymore. And once you know the Green's function, you know all the basic thermodynamic properties of the system. This has never been done before."

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2 / 5 (4) Mar 18, 2012
Not much is written about the validating experiment: was this about predicting or simulating the transition temperature of a particular superconducting material?
1 / 5 (5) Mar 18, 2012
Nope, none such experiments exist. The experiments are expensive today whereas the computers are everywhere. And physicists need to write publications all the time to keep their grant and salaries.
3 / 5 (2) Mar 19, 2012
I'm pretty sure this is the paper
arXiv 1110.3747v1 [cond.mat-quant.gas] 17 Oct 2011

The experiment is in the paper.
From the paper:
In short, ultracold fermionic 6Li is brought to
degeneracy via sympathetic cooling with 23Na. A two state
mixture of the two lowest hypefine states of 6Li
is further cooled in a hybrid magnetic and optical trap
at the broad Feshbach resonance at 834 G. We employ
high-resolution in situ absorption imaging to obtain the
column density of the gas, that is converted into the full
3D density via the inverse Abel transform [28]. .... Equipotential averaging yields low-noise profiles of density N versus potential V . Density is absolutely calibrated by imaging a highly degenerate, highly imbalanced Fermi mixture, and fitting
the majority density profile to the ideal Fermi gas
EOS [24]. In contrast to previous studies [22, 23], our
analysis does not rely on the assumption of harmonic

I wonder what the GRACE Collaboration will make of this?
2.3 / 5 (3) Mar 19, 2012
I wonder what the GRACE Collaboration will make of this?

Gravity Recovery and Climate Experiment?
5 / 5 (1) Mar 19, 2012
This technique seems like it could have lots of really good uses! I look forward to seeing if it can provide some insights to the strongly interacting theory of QCD.

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