# 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. 