Abstract

We calculate the universal ratios R2k of renormalized coupling constants g2k entering the critical equation of state for the generalized Heisenberg (three-dimensional n-vector) model. Renormalization group (RG) expansions of R8 and R10 for arbitrary n are found in the four-loop and three-loop approximations respectively. Universal octic coupling R8⁎ is estimated for physical values of spin dimensionality n=0,1,2,3 and for n=4,...64 to get an idea about asymptotic behavior of R8⁎. Its numerical values are obtained by means of the resummation of the RG series and within the pseudo-ε expansion approach. Regarding R10 our calculations show that three-loop RG and pseudo-ε expansions possess big and rapidly growing coefficients for physical values of n what prevents getting fair numerical estimates.

Highlights

  • The thermodynamics of O(n)-symmetric systems near the Curie point is characterized by universal parameters that depend only on the ordering field dimensionality n and the number of space dimensions D

  • We extend the Renormalization group (RG) expansion of effective coupling constants g8 up to four-loop order and calculate three-loop RG series for g10 under arbitrary n

  • Since the equation of state and expressions for nonlinear susceptibilities contain the universal ratios rather than effective coupling constants themselves it is reasonable to have the pseudo-ε expansions for the critical values of these ratios

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Summary

Introduction

The thermodynamics of O(n)-symmetric systems near the Curie point is characterized by universal parameters that depend only on the ordering field dimensionality n and the number of space dimensions D. Estimating universal critical values of quartic and sextic effective couplings by means of the field-theoretical renormalization group (RG) approach has shown that RG technique enables one to get rather accurate numerical results for these quantities. Since the equation of state and expressions for nonlinear susceptibilities contain the universal ratios rather than effective coupling constants themselves it is reasonable to have the pseudo-ε expansions for the critical values of these ratios. These RG expansions and τ-series will be used to estimate higher-order effective couplings near the Curie point

Wilson fixed point coordinate
Fitting
Octic effective interaction at criticality
Tenth-order coupling in the critical region
Conclusion
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