Examples of overlapping convergent expansions of scaling variables

TL;DR

This paper constructs and analyzes series expansions of RG scaling variables in Dyson's hierarchical model, demonstrating their overlapping convergence domains and enabling accurate analytical expressions across critical and high-temperature regions.

Contribution

It introduces a method to construct and analyze overlapping convergent series expansions of RG scaling variables in hierarchical models.

Findings

Scaling variables have overlapping convergence domains.

Series expansions accurately describe critical and high-temperature behavior.

Quantities like magnetic susceptibility can be expressed in these variables.

Abstract

We construct series expansions for the scaling variables (which transform multiplicatively under a renormalization group (RG) transformation) in examples where the RG flows, going from an unstable (Wilson's) fixed point to a stable (high-temperature) fixed point, can be calculated numerically. The examples are Dyson's hierarchical model and a simplified version of it. We provide numerical evidence that the scaling variables about the two fixed points have overlapping domain of convergence. We show how quantities such as the magnetic susceptibility can be expressed in terms of these variables. This procedure provide accurate analytical expressions both in the critical and high-temperature region.