Stability of a cubic fixed point in three dimensions. Critical exponents for generic N

TL;DR

This paper analyzes the stability and critical exponents of the cubic fixed point in three-dimensional models with isotropic and cubic interactions, using high-order perturbative RG calculations and resummation techniques.

Contribution

It provides the first detailed four-loop RG analysis in 3D for arbitrary N, establishing the stability of the cubic fixed point for N≥3 and calculating accurate critical exponents.

Findings

Critical dimensionality N_c=2.89 ± 0.02

Cubic fixed point stable for N≥3 in 3D

Precise critical exponents for phase transitions

Abstract

The detailed analysis of the global structure of the renormalization-group (RG) flow diagram for a model with isotropic and cubic interactions is carried out in the framework of the massive field theory directly in three dimensions (3D) within an assumption of isotropic exchange. Perturbative expansions for RG functions are calculated for arbitrary $N$ up to the four-loop order and resummed by means of the generalized Pad$\acute{\rm e}$-Borel-Leroy technique. Coordinates and stability matrix eigenvalues for the cubic fixed point are found under the optimal value of the transformation parameter. Critical dimensionality of the model is proved to be equal to $N_c=2.89 \pm 0.02$ that agrees well with the estimate obtained on the basis of the five-loop $\ve$-expansion [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B342, 284 (1995)] resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for $N\ge3$, and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when N=0 is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.