Renormalization Group Functions for Two-Dimensional Phase Transitions: To the Problem of Singular Contributions

TL;DR

This paper investigates the discrepancy between RG-derived and exact critical exponents in 2D phase transitions, showing that accurate exponents can be obtained without assuming singular contributions, using a new summation algorithm.

Contribution

Introduces a new algorithm for summing divergent series that demonstrates exact critical exponents can be achieved without nonanalytic RG contributions.

Findings

Exact critical exponents can be obtained with reasonable coefficient functions.

Small nonmonotonities in coefficient functions are sufficient for accurate results.

Singular contributions in RG functions are not necessary for explaining the exponents.

Abstract

According to the available publications, the field theoretical renormalization group (RG) approach in the two-dimensional case gives the critical exponents that differ from the known exact values. This fact was attempted to explain by the existence of nonanalytic contributions in the RG functions. The situation is analysed in this work using a new algorithm for summing divergent series that makes it possible to analyse dependence of the results for the critical exponents on the expansion coefficients for RG functions. It has been shown that the exact values of all the exponents can be obtained with a reasonable form of the coefficient functions. These functions have small nonmonotonities or inflections, which are poorly reproduced in natural interpolations. It is not necessary to assume the existence of singular contributions in RG functions.