A Hermite-Padé perspective on the renormalization group, with an application to the correlation function of Lieb-Liniger gas

Renormalization group (RG) methods are based on symmetry principles that are peculiar to quantum field theory and certain related settings. However, in cases when the β-function depends only on the coupling, the RG technology does include a part-which is related to the equation for the running coupling and which we will call the 'single variable RG' (SVRG)-that turns out to be a purely formal method for improving the convergence properties of series expansions. In fact, we show that SVRG is an integral Hermite-Padé approximation preceded by a series reversion. A complementary viewpoint on what SVRG does to an expansion is that it performs a 'minimal upgrade to a self-similar form', the minimality being crucial. In applications, SVRG is especially useful in problems in which one knows not only the first few terms of an expansion valid for small values of a variable, but also the exponent (but not the prefactor) of the scaling law for large values of the variable. As a practical example, we extract the scaling-law prefactor for the one-body density matrix of the Lieb-Liniger gas. Using a new result for the fourth-order term in the short-distance expansion, we find remarkable agreement with known quantum diffusion Monte Carlo numerical results.