The stability of a cubic fixed point in three dimensions from the renormalization group

Abstract

The global structure of the renormalization-group flows of a model with isotropic and cubic interactions is studied using the massive field theory directly in three dimensions. The four-loop expansions of the β-functions are calculated for arbitrary N. The critical dimensionality N c = 2.89 ± 0.02 and the stability matrix eigenvalues estimates obtained on the basis of the generalized Padé-Borel-Leroy resummation technique are shown to be in a good agreement with those found recently by exploiting the five-loop ε-expansions.

Randomly dilute Ising model: A nonperturbative approach

“…Considering the cubic anisotropy, we have determined an estimate of N c above which the cubic fixed point is stable and found N c (d = 3) = 3.1. This value differs significantly from those obtained from recent perturbative approaches for which N c < 3 10,11,14,15,16,17,18 . For instance, in a recent six-loop calculation, N c is found to be equal to 2.89(4) 15 , a value already obtained from lower order computations 16,17 .…”

Critical behavior of certain antiferromagnets with complicated ordering: Four-loopɛ-expansion analysis

“…From the recent pseudo-ε expansion analysis of the real hypercubic model [35] one can extract N c = 1.431(3). However the most accurate estimate N c = 1.445 (20) results from the value n c = 2.89(4) obtained on the basis of the numerical analysis of the four-loop [34] and the show that the resummation method employed in this paper gives the same results as the conventional technique using the exact values of the asymptotic parameters (see Refs. [27] and [28]) 9 The fixed point is called to be of the "saddle" type provided their eigenvalue exponents ω 1 and ω 2 are of opposite sings at the (u, v) plane.…”

“…Resummation of that series gave the estimate n c = 2.894(40) (see Ref. [34]). Therefore we conclude that N c = 1.447 (20) from the five-loops.…”

Critical thermodynamics of three-dimensionalMN-component field model with cubic anisotropy from higher-loop ε expansion

“…Resummation of that series gave the estimate N R c = 2.894(40) [9]. Therefore we conclude that N (25) follows from a constrained analysis of N R c taking into account N R c = 2 in two dimensions [10].…”

“…In this limit the Ising critical exponents take Fisher-renormalized values [6]. Since static critical phenomena in a cubic crystal as well as in randomly diluted Ising spin systems are well understood [7,8,9,10,11] 2 , we will focus here on the critical behaviour of the above mentioned multisublattice antiferromagnets. This is the case of M = 2 and N = 2, N = 3 in the fluctuation Hamiltonian (1).…”