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High-precision calculations in quantum statistical physics from a divergent sum

What is the common point between the behaviors of nucleons in nuclei and of electrons in materials such as high-temperature superconductors ? The difficulty to compute these behaviors reliably and accurately, due to the strong correlations between the particles, which belong to the fermion family. To test different theoretical approaches, physicists compare calculations with experiments on gases of atoms at temperatures very close to absolute zero, in particular in the maximally interacting regime called “unitary Fermi gas”. Fifteen years after such a gas was first produced, experimental progress enable increasingly accurate measurements of various properties, notably the probability for two atoms to be very close to each other, determined by a parameter called “contact”. Existing calculations of the contact were spread over a range of 60%, each of the employed methods being affected by poorly controlled approximations. New values of the contact with a controlled precision of 2% are now published in Physical Review Letters by researchers from Laboratoire de Physique Statistique and Laboratoire Kastler Brossel with their international collaborators. These results agree with experimental data. The employed computational method may be applicable to many other systems of strongly correlated fermions.

Two fermions can pair up to form a pair, and this pair can in turn dissociate into two fermions. When two fermions are separated, one of them can pair up with a third fermion, forming a second pair. To compute the value of the contact, the physicists have added up the contributions of about one hundred thousand such processes, involving up to nine pairs. But the terms of this sum increase faster than exponentially, and for such a strong divergence, it is not always possible to give a meaning and to attribute a unique value to the sum. Riccardo Rossi and coworkers were able to show, using field theory methods, that this is indeed possible for the unitary Fermi gas, by constructing and analyzing a function standing behind the sum. This was the subject of another article published in the same volume of Physical Review Letters. Part of the work was to rediscover a theorem from 1919 that was hidden in the voluminous literature on the subject. As is ofter the case in physics, some stages of the reasoning are conjectures without rigorous mathematical proof, which reinforces the importance of comparisons with experiments, beginning with more accurate measurements of the contact currently underway in several laboratories.

<a href='http://archive.lps.ens.fr/spip.php?article3767'>High-precision calculations in quantum statistical physics from a divergent sum</a>

Figure : Example of Feynman diagram taken into account in the calculation. The arrows represent fermions, and the rectangle represent pairs of fermions. These pairs are bosons. This representation is particularly suited for the unitary Fermi gas, which is in the middle of the cross-over between purely fermionic physics (at weak attraction between fermions) and purely bosonic physics (at strong attraction, two fermions forming a tightly bound pair).

More :
R. Rossi et al., Contact and Momentum Distribution of the Unitary Fermi Gas, Physical Review Letters 121, 130406 (2018)
R. Rossi et al., Resummation of diagrammatic series with zero convergence radius for strongly correlated fermions, Physical Review Letters 121, 130405 (2018)

Author affiliations :
Laboratoire de Physique Statistique (ENS / CNRS / Sorbonne Université / Université Paris Diderot)
Laboratoire Kastler Brossel (ENS / CNRS / Sorbonne Université / Collège de France)
University of Tokyo, Department of Applied Physics
University of Massachusetts, Amherst, Department of Physics
King’s College, London, Physics Department

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