Critical properties of random anisotropy magnets

Abstract

The problem of critical behaviour of three dimensional random anisotropy magnets, which constitute a wide class of disordered magnets is considered. Previous results obtained in experiments, by Monte Carlo simulations and within different theoretical approaches give evidence for a second order phase transition for anisotropic distributions of the local anisotropy axes, while for the case of isotropic distribution such transition is absent. This outcome is described by renormalization group in its field theoretical variant on the basis of the random anisotropy model. Considerable attention is paid to the investigation of the effective critical behaviour which explains the observation of different behaviour in the same universality class.Comment: 41 pages, 10 figure

Critical slowing down in random anisotropy magnets

“…Therefore, the m-vector magnets with cubic random axis distribution belong to the universality class of the random-site Ising magnets. The non-asymptotic critical behaviour of the RAM essentially differs from that of the random-site model as was demonstrated in statics in [15]. The same concerns the non-asymptotic dynamical critical behaviour: the critical slowing down in RAM is governed by z eff exponent as explained below.…”

“…Within the massive renormalization they have been calculated already in five-loop [30] approximation. Calculating the dynamical function ζ Γ within two-loop order we use the static RG functions of the same order [15]. Since the series for these static functions are known to be asymptotic at best we use Padé-Borel resummation scheme [36] described in detail in [15].…”

“…Calculating the dynamical function ζ Γ within two-loop order we use the static RG functions of the same order [15]. Since the series for these static functions are known to be asymptotic at best we use Padé-Borel resummation scheme [36] described in detail in [15]. The FPs values and solutions of flow equations [15] are obtained based on these functions.…”

“…The region of physical importance u > 0, v > 0, w < 0 includes 10 FPs. Below we list the most interesting FPs together with the asymptotic value for the z exponent (for the numerical values of the FPs coordinates obtained in two-loop approximation using the Padé-Borel resummation see table 2 of [15], the value of the z exponent, however, is calculated by a direct substitution of FP coordinates into equation (13) Here, we keep the FP numbering of [15,26,27,29,30]. From the above list, only the random Ising (XV) and polymer (III) FPs are stable.…”

“…With the Lagrangian (8) depending on the bare quantities (denoted by the suband superscripts "o") we analyze within the field theory [3] the dynamical vertex functions. As far as the static RG functions have been obtained before [15,29], we need only to calculate the two-point dynamical vertex functionΓ i,j ϕϕ (r 0 , {u i,0 },Γ, k, ω) =Γφ ϕ (r 0 , {u i,0 },Γ, k, ω)δ i,j . The calculations are performed using Feynman diagrams, elements for them are given in figure 1 whereas the one-and two-loop contributions toΓ i,j ϕϕ are depicted in figure 2.…”