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Physics at the edge of the cut

October 14, 2015 - Perturbation theory (PT) is an essential tool for the mathematical modeling of physical systems. It allows one to approximate the solution of a complex problem in terms of the solution of a much simpler unperturbed one. PT can be used to find approximations to almost  anything and it is used across fields in the physical and engineering sciences: the deflection of a wing and the magnetic moment of nuclei furnish examples from an endless list of possible applications.

In PT physical quantities are approximated as series expansions, in essence sums of infinitely many terms;  the first, zeroth order, term in the series corresponds to the unperturbed value, while higher-order terms correspond to corrections due to the perturbation. For PT to  be predictive the sum has to converge; the first-order correction needs to be small, the second much smaller, and so on so forth. In such cases  the sum of all terms yields a finite number that accurately represents the experimentally-observed values of physical quantities. 

In many cases, however, it turns out that  PT predicts infinite values for the physical observables: higher-order terms in the series can be in fact much larger than lower-order ones and the sum is infinite. In such cases PT is said to be divergent. 

Remarkably, physicists and mathematicians know ways to make sense of the infinities of divergent PT. They use so-called resummation techniques; these assign, for example, the number -1/12 to the infinite sum 1+2+3+4+... Divergent series and resummation techniques  can be highly counter-intuitive and have a reputation of being somewhat of an obscure art;  Abel once wrote  that divergent series are “the work of the Devil”, and lead to “disasters and paradoxes”. Despite these historical concerns modern resummation approaches can work well in many cases, managing to transform the unphysical infinities from divergent PT  into finite numbers that are often in good agreement with experimental data. Unfortunately these approaches typically need the calculation of very high-order corrections, thereby increasing the computational cost to prohibitive levels. By being too computationally demanding standard resummation methods such as Pade or Borel-Pade have  limited applicability.

Now in a recent paper entitled Nonperturbative Quantum Physics from Low-Order Perturbation Theory and published in the prestigious Physical Review Letters, DPA researchers Dr. Hector Mera and Prof. Branislav K. Nikolic in collaboration with Prof. Thomas G. Pedersen from Aalborg University in Denmark, have put forward a novel hypergeometric resummation method that is computationally simple and yields excellent approximations where standard resummation  approaches fail.  The divergence in PT is associated to the presence of a singularity in an abstract complex-plane of perturbations. Standard resummation approaches are well suited to model one type of singularities called poles but have a hard time mimicking the other main type of singularity, called a branch cut. A cut can be thought of as the limit of infinitely-many poles sitting infinitesimally close to each other.  Hypergeometric resummation exploits the flexibility of hypergeometric functions to mimic both types of singularities and are highly accurate but much simpler than traditional resummation methods.

In more recent work the team is applying their method to a variety of problems: electron-phonon scattering in molecular junctions, Stark effect and dopant de-activation in quantum confined systems, the physics of strongly correlated electrons as well as generalizing and improving hypergeometric resummation.


For more informatrion about theoretical and computational condensed matter and nanophysics reseach conducted by Nikolic group visit their Website.


Physical Review Letters (PRL) is the world’s premier physics letter journal. It publishes short, high quality reports of significant and notable results in the full arc of fundamental and interdisciplinary physics research. PRL provides readers with the most influential developments and transformative ideas in physics with the goal of moving physics forward. PRL is the most cited physics journal — every two minutes someone cites a PRL. Authors gain high visibility, rapid publication, and broad dissemination of their work.  PRL's 2014 impact factor is 7.512, according to the 2014 Journal Citation Reports Science Edition (Thomson Reuters, 2015). The 2014 impact factor represents the average number of citations received in 2014 for papers published in 2013 and 2012. A more detailed explanation of impact factors can be found on the Thomson Reuters web site. The 2014 5-year impact factor, the average number of times articles published in 2009-2013 have been cited in 2014, is 7.360, and the cited half-life, the median age of articles cited in 2014, is 8.8 years. PRL's 2014 immediacy index, the average number of citations in 2014 to papers published in that year, was 2.531. PRL is ranked first among physics and mathematics journals by Google Scholar using their five-year h-index metrics, and first among physics journals with an Eigenfactor of 0.94. The number of articles published in PRL in 2014 was 2,846, with a total of 14,660 pages. The current projections for 2015 are 2,630 articles and 13,580 pages.