Small numerators canceling small denominators of the high-temperature scaling variables: a systematic explanation in arbitrary dimensions

TL;DR

The paper presents a systematic method to express susceptibility and derivatives in terms of high-temperature scaling variables in the Dyson hierarchical model, demonstrating cancellation of small denominators and implications for phase existence across dimensions.

Contribution

It provides a closed-form solution for the linearized problem and shows how to handle small denominator issues in arbitrary dimensions, extending understanding of RG flows and critical phenomena.

Findings

All poles up to order 7 are canceled by zeroes in positive dimensions.

Continuity in dimension helps resolve ambiguities in fixed-dimension calculations.

Small denominator problems persist at negative dimensions, affecting finite-size corrections.

Abstract

We describe a method to express the susceptibility and higher derivatives of the free energy in terms of the scaling variables (Wegner's nonlinear scaling fields) associated with the high-temperature (HT) fixed point of Dyson hierarchical model in arbitrary dimensions. We give a closed form solution of the linearized problem. We check that up to order 7 in the HT expansion, all the poles ("small denominators") that would naively appear in some positive dimension are canceled by zeroes ("small numerators"). The requirement of continuity in the dimension can be used to lift ambiguities which appear in calculations at fixed dimension. We show that the existence of a HT phase in the infinite volume limit for a continuous set of values of the dimension, requires that this mechanism works to all orders. On the other hand, most poles at negative values of the dimensional parameter (where the free energy density is not well-defined, but RG flows can be studied) persist and reflect the fact that for special negative values of the dimension, finite-size corrections become leading terms. We show that the inverse problem is also free of small denominator problems and that the initial values of the scaling variables can be expressed in terms of the infinite volume limit of the susceptibility and higher derivatives of the free energy. We discuss the existence of an infinite number of conserved quantities (RG invariants) and their relevance for the calculation of universal ratios of critical amplitudes.