Universal effective coupling constants for the generalized Heisenberg model

TL;DR

This paper calculates universal critical coupling constants for the generalized Heisenberg model, providing refined numerical values for different component numbers and assessing the accuracy of various approximation methods.

Contribution

It introduces a method to determine universal coupling constants $g_4$ and $g_6$ for Heisenberg ferromagnets using high-order loop calculations and resummation techniques.

Findings

Calculated $g_6^*$ values with less than 1.6% error.

Compared results with $1/n$ method to estimate its accuracy.

Provided universal constants for different $n$-component systems.

Abstract

The aim of this study is to find universal critical values of the dimensionless effective coupling constant $g_6$ and refined universal values $g_4$ for Heisenberg ferromagnets with $n$-component order parameters. These constants appear in the equation of state and determine the nonlinear susceptibilities $\chi_4$ and $\chi_6$ in the critical region. The sextic coupling $g_6$ is calculated as a function of $g_4$ in the three-loop approximation, the series obtained is resummed by the Pade-Borel method and then by substituting the Wilson fixed point coordinate $g_4^*$ into resultant expression numerical values of $g_6^*$ are obtained for different $n$. The numbers $g_4^*$ were determined from the six-loop expansions for $\beta$ function resummed using Borel-transformation-based techniques. The analysis of the accuracy of these $g_6^*$ values showed that they differ from the true values by no more than 1.6 %. The values of $g_6^*$ thus found were compared also with those given by the $1/n$ method what allowed to estimate the level of accuracy of this method.