Hypergeometric continuation of divergent perturbation series. II. Comparison with Shanks transformation and Pad\'e approximation

TL;DR

This paper demonstrates that hypergeometric continuation is an effective and reliable method for analyzing divergent perturbation series in quantum phase transitions, outperforming traditional techniques like Shanks and Padé in certain aspects.

Contribution

It introduces hypergeometric continuation as a superior alternative for analyzing divergent series and extracting critical exponents in quantum phase transition studies.

Findings

Hypergeometric continuation accurately reproduces phase diagrams.

It allows determination of divergence exponents at transition points.

Compared to Shanks and Padé, it offers improved reliability and efficiency.

Abstract

We explore in detail how analytic continuation of divergent perturbation series by generalized hypergeometric functions is achieved in practice. Using the example of strong-coupling perturbation series provided by the two-dimensional Bose-Hubbard model, we compare hypergeometric continuation to Shanks and Pad\'e techniques, and demonstrate that the former yields a powerful, efficient and reliable alternative for computing the phase diagram of the Mott insulator-to-superfluid transition. In contrast to Shanks transformations and Pad\'e approximations, hypergeometric continuation also allows us to determine the exponents which characterize the divergence of correlation functions at the transition points. Therefore, hypergeometric continuation constitutes a promising tool for the study of quantum phase transitions.