Universal effective coupling constants for the generalized Heisenberg model

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 $g_4^{\ast }$ into resultant expression numerical values of $g_6^{\ast }$ are obtained for different $n$. The numbers $g_4^{\ast }$ were determined from the six-loop expansions for $\beta$ function resummed using Borel-transformation-based techniques. The analysis of the accuracy of these $g_6^{\ast }$ values showed that they differ from the true values by no more than 1.6 %. The values of $g_6^{\ast }$ thus found were compared also with those given by the $1/n$ method what allowed to estimate the level of accuracy of this method.

Comments: 6 pages, 2 figures, 1 table