The Analytic Renormalization Group

TL;DR

This paper introduces the Analytic Renormalization Group, a set of universal linear equations derived from analyticity that relate low and high energy Fourier coefficients of finite temperature correlators, enabling improved data analysis.

Contribution

It formulates model-independent linear constraints on Fourier coefficients of finite temperature correlators and develops an algorithm to systematically refine approximate data sets using these constraints.

Findings

Universal linear equations relate Fourier coefficients across energy scales.

The method improves the accuracy of Monte Carlo simulation data.

Explicit examples demonstrate the effectiveness of the constraints.

Abstract

Finite temperature Euclidean two-point functions in quantum mechanics or quantum field theory are characterized by a discrete set of Fourier coefficients $G_{k}$, $k\in\mathbb Z$, associated with the Matsubara frequencies $\nu_{k}=2\pi k/\beta$. We show that analyticity implies that the coefficients $G_{k}$ must satisfy an infinite number of model-independent linear equations that we write down explicitly. In particular, we construct "Analytic Renormalization Group" linear maps $\mathsf A_{\mu}$ which, for any choice of cut-off $\mu$, allow to express the low energy Fourier coefficients for $|\nu_{k}|<\mu$ (with the possible exception of the zero mode $G_{0}$), together with the real-time correlators and spectral functions, in terms of the high energy Fourier coefficients for $|\nu_{k}|\geq\mu$. Operating a simple numerical algorithm, we show that the exact universal linear constraints on $G_{k}$ can be used to systematically improve any random approximate data set obtained, for example, from Monte-Carlo simulations. Our results are illustrated on several explicit examples.