Critical slowing down in random anisotropy magnets

We study the purely relaxational critical dynamics with non-conserved order parameter (model A critical dynamics) for three-dimensional magnets with disorder in a form of the random anisotropy axis. For the random axis anisotropic distribution, the static asymptotic critical behaviour coincides with that of random site Ising systems. Therefore the asymptotic critical dynamics is governed by the dynamical exponent of the random Ising model. However, the disorder influences considerably the dynamical behaviour in the non-asymptotic regime. We perform a field-theoretical renormalization group analysis within the minimal subtraction scheme in two-loop approximation to investigate asymptotic and effective critical dynamics of random anisotropy systems. The results demonstrate the non-monotonic behaviour of the dynamical effective critical exponent $z_{\rm eff}$.


Introduction
The concept of scaling plays a central role in modern theory of critical phenomena [ 1]. Introducing a set of appropriate scaling variables a large amount of experimental and numerical data can be described by a few scaling functions [ 2]. The most prominent effect in dynamical critical phenomena is critical slowing down which consists in an increase of the relaxation time approaching the critical point. This is induced by the divergence of the correlation length ξ at the critical point and causes also relaxation time τ to diverge with the dynamical critical exponent z: Renormalization group (RG) theory, with its concepts of invariance of the system at the critical point against changes in length scale gives a basis to universality in connection with fixed points and static [ 3] and dynamic scaling [ 4]. This holds however only in the asymptotic region in the vicinity of the critical point. Further away scaling breaks down and the description of critical phenomena becomes more complicated and involves non-universal characteristics both in statics and in dynamics [ 1,5].
Another important point for observing the behaviour in certain universality classes is the homogeneity of the system under consideration. Therefore the effect of impurities on critical behaviour is of considerable interest. However it turned out the disordered systems may show also scaling within a certain universality class. This might be the universality class of a pure system or a new one [ 6,7,8,9]. Moreover the changes introduced by the disorder depend on the type of this disorder; namely is it introduced by dilution (random site [ 10] or random bond [ 11] systems), or as a random field [ 12], random connectivity [ 9,13] or as an anisotropy [ 14]. The defects may be correlated [ 7,8] or not. It may even happen that the second order transition of the pure system is destroyed [ 16,17].
It further turned out that considering a specific system with defects the behaviour near the critical point seems to be non-universal. Knowing that the non-asymptotic behaviour is non-universal it became necessary to study the non-asymptotic behaviour of such systems in more detail. Indeed in many systems (e.g with site disorder) effective critical behaviour can explain the experimental situation [ 11,18].
RG investigations of dynamic critical behaviour is in many cases technically much more involved in comparison to statics. As a consequence results for dynamics are known and in much lower approximations (in most cases only up to two loop order). On the other hand for the dynamics of magnetic systems with mode coupling terms the dynamical critical exponent is known exactly or contains only a static exponent.
In this paper, we will present an analysis of the dynamical critical behaviour of random anisotropy magnets which constitute a large class of disordered systems [ 14]. In order to give some examples, the majority of the amorphous rare-earth containing alloys is recognized as random anisotropy magnets [ 14], certain crystalline compounds with an rear-earth component belong to this class too [ 19]. Random anisotropy characterizes also molecular based magnets [ 20], nanocrystalline materials [ 21], as well as granular systems [ 22]. Moreover, the analysis involving random anisotropy found its application also in the interpretation of the phase transition in liquid crystals in porous media [ 23].
The model currently used for the description of random anisotropy systems was introduced in the early 70-ies by Harris, Plischke, and Zuckermann [ 24]. It describes m-component spins located on the sites of a d-dimensional lattice, each spin subjected to a local anisotropy of random orientation. The Hamiltonian of the random anisotropy model (RAM) reads: Here, S R = (S 1 R , . . . , S m R ), vectors R span sites of a d-dimensional cubic lattice, D 0 > 0 is an anisotropy constant,x is a random unit vector specifying direction of the local anisotropy axis. The interaction J R,R ′ is assumed to be ferromagnetic. Note, that for the Ising-like magnets, m = 1, the last term in (2) is just a constant, therefore the random anisotropy is present for m > 1 only. Below, we will consider quenched disorder, when the vectorsx R in (2) are randomly distributed with a distribution function p(x) and fixed in a certain configuration. It is well established by now, that the anisotropy axis distribution plays a crucial role for the origin of the low-temperature phase in the RAM. In particular, when the random vectorsx R point with equal probability towards any direction, such a distribution may be called an isotropic one, the ferromagnetic ordering is impossible [ 25] for spatial dimension d ≤ 4. Whereas it may occur for an anisotropic distribution. In statics, this situation was corroborated by the RG studies of RAM [ 26,27,28,29,30] restrictingx to point along one of the 2m directions of the axeŝ k i of a hypercubic lattice (cubic distribution): with Kronecker deltas δ(y). In this case, the second order phase transition into the magnetically ordered low-temperature phase occurs and in asymptotics it is characterized by the critical exponents of the random-site Ising model, the fact suggested already in Ref. [ 27] and confirmed later in Refs. [ 28,29,30].
Taken that the studies of static criticality of random anisotropy magnets are far from being as intensive as those of the diluted magnets [ 10], even less is known about their dynamic critical behaviour. The dynamical models for systems with isotropic distribution of local anisotropy axis were considered in Refs. [ 31,32,33], the first order RG calculations were performed in Ref. [ 34]. Whereas the problem of dynamic critical behaviour of a RAM with an anisotropic random axis distribution as to our knowledge up to now remained untouched.
In this paper, we consider a purely relaxational dynamics of a three-dimensional (d = 3) RAM with non-conserved order parameter (model A in classification of Ref. [ 4]) and a cubic random axis distribution (3). As far as the static critical behaviour of such magnets (note, with m > 1) belongs to the universality class of a random-site Ising model [ 27,28,29,30], for which the heat capacity does not diverge [ 10]. Therefore, the critical dynamics of such a model is governed in asymptotics by the model A random-site Ising magnet dynamical critical exponent for any order parameter components number m. However in the non-asymptotic region, the model possesses a rich effective critical behaviour, as will be shown by our subsequent analysis. Taken that it is this effective behaviour which is observed both in experiments and in the MC simulations, it is important to have a RG prediction for typical scenarios of the approach to criticality in random anisotropy magnets.
The rest of the paper is organized as following: in the next section 2 we present the Langevin equations governing model A dynamics and describe the renormalization procedure, in section 3 we give results of our calculations obtained in two-loop approximation and display possible scenarios of effective critical behaviour. Conclusions and outlook are given in section 4.

Model equations and renormalization
We consider the dynamics for model (2) with random axis distribution (3) to be relaxational with non-conserved m-component order parameter ϕ 0 ≡ ϕ 0 (R). In this case the relaxation of the order parameter is described by the Langevin equation: with the Onsager coefficientΓ, stochastic forces θ ϕ i obeying the Einstein relations: and the disorder-dependent equilibrium effective Hamiltonian H: In (6), the field ϕ 0 is an m-component vector, D is proportional to the anisotropy constant of the spin Hamiltonian (2) with D 0 ;r and v 0 are defined by D 0 and the fourth order coupling of the m-vector magnet (see Ref. [ 29] for details). We treat the critical dynamics of the disordered model within the field theoretical RG method based on the Bausch-Janssen-Wagner formulation [ 35], where the appropriate Lagrangians are studied. For the model equations (4)-(5) the Lagrangian reads: with a new auxiliary response m-component field˜ ϕ 0 . Here and below sums over field components span values from 1 to m. Studying critical properties of disordered systems one should average over the random degrees of freedom. In order to treat quenched disorder, in statics often the replica trick is used [ 10]. However, it was established in [ 32] that in dynamics it is not necessary to make use of the replica trick: it is sufficient to the average over the random variablesx with their distribution (3). Then the Lagrangian for the model reads: Here, r 0 is proportional to the temperature distance to the mean field critical point and the bare couplings are u 0 > 0, v 0 > 0, w 0 < 0. Moreover, u 0 and w 0 are connected to the moments of the distribution (3) in such a way that w 0 /u 0 = −m.
The term with y 0 does not appear after averaging over disorder, however it should be added since it will be generated within the renormalization procedure. The set of static couplings {u 0 , v 0 , w 0 , y 0 } will be denoted below by {u 0,i }, i = 1, .., 4.
With the Lagrangian (8) depending on the bare quantities (denoted by the suband superscripts 'o') we analyze within 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 the Fig. 1 whereas the one-and two-loop contributions toΓ i,j ϕϕ are depicted in Fig. 2. We perform renormalization ofΓφ ϕ within the minimal subtraction scheme [ 3]. In this scheme, in order to define renormalized static (ϕ, r, {u i }) and dynamic (φ, Γ) fields and couplings, the renormalization factors Z ϕ , Z r , Z u i and Zφ, Z Γ are introduced by: Here, µ is the external momenta scale, ε = 4 − d, and A d is a geometrical factor. The critical behaviour of the system is described by the following RG functions: where the symbol 0 means differentiation at fixed bare parameters. The β-functions determine the RG flow of couplings under renormalization: and the flow parameter ℓ is related to the distance from the critical point. Subsequently, an information about the critical behaviour of a system can be obtained from analysis of the fixed points (FPs) of the flow equations (11). A FP {u * i } is defined as simultaneous zero of all β-functions: The stable and from initial conditions accessible FP corresponds to the critical point of the system. A FP is stable if all eigenvalues ω i of the stability matrix B i,j = ∂βu i ∂u j calculated at this FP have positive real parts.
The FP values of the RG ζ-functions (10) determine the asymptotic values of critical exponents. In particular, the dynamical asymptotic critical exponent z (1) is given at the stable and accessible FP by: While the effective dynamical exponent z eff is calculated in the non-asymptotical region, where the renormalized couplings did not reach their FP values and it is defined by the solutions of the flow equations (11): We neglect contributions to z eff coming from the amplitude function because they are considered to be small.

Results
The static RG functions within the minimal subtraction scheme are known in two-loop [ 15] approximation. 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 series for these static functions are known to be asymptotic at best we use Padé-Borel resummation scheme [ 36] in details described in Ref. [ 15]. The FPs values and solutions of flow equations [ 15] obtained on the basis of these functions we use in present study adding to them the new two-loop expression for the function ζ Γ . The last is derived from the vertex functionΓ i,j ϕϕ discussed in Section 2 and reads: (m + 2)uv + 5u 2 + 5w 2 + 10uw + 3yw + 3vw + 3uy .
Here, we keep the FP numbering of Refs. [ 15,26,27,29,30]. From the above list, only the random Ising (XV) and polymer (III) FPs are stable. However the polymer FP is not accessible from the physical initial conditions. This leads to the conclusion [ 27,28,29,30] that the random Ising FP XV governs the critical behaviour. 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 differs essentially from that of the random-site model as was demonstrated in statics in Ref. [ 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.
The crossovers between different FPs lead to a rich pictures of possible RG flows [ 15]. Many flows are influenced by the Ising FPs V and X. Introducing into (14) several typical RG flows which start from the physical region of initial couplings one obtains different regimes for approach of the effective dynamical exponent z eff to asymptotics. The dependence on the flow parameter ℓ of z eff for easy-plane (m = 2) and Heisenberg (m = 3) magnets are shown in the Figs. 3 and 4 correspondingly. Although the asymptotic exponents of the random anisotropy magnets considered here are the same as those of the random-site (diluted) Ising magnets, the approach to the asymptotical region essentially differs from the diluted magnets. It is defined by smallest static stability exponent ω = −0.0036 [ 15] which is equal in absolute value to the ratio of heat capacity critical exponent α r and correlations length critical exponent ν r of random-site Ising model [ 30]. As a consequence the Wegner correction to scaling is ∆ = ων r = −α r . The high-loop estimate gives ∆ ≈ 0.049 ± 0.009 [ 30]. Such a small value of ∆ means that the approach to the asymptotic values is very slow. Therefore practically only the non-asymptotic critical behaviour governed by effective critical exponents will be observed experimentally or in the numerical simulations.
As it is seen from Figs. 3, 4, another particular feature of z eff seems to be that it reaches its asymptotic value z always from the region z eff > z. Therefore, in an experimental situation, a decrease of z eff may serve as an evidence of approach to the asymptotics. Note that such a scenario is an intrinsic feature of the critical slowing down in random-anisotropy magnets. When disorder is implemented by dilution of the non-magnetic component, an approach of z eff to its asymptotic value is not necessarily only from above [ 38].

Conclusions
In this paper, we have analyzed the critical slowing down in magnets influenced by random anisotropy. These magnets have a second order phase transition to ferro- magnetic order for an anisotropic (cubic) random axis distribution. Therefore, our goal was to study relaxational dynamics of the non-conserved order parameter in the vicinity of the phase transition point. For this purpose we completed previous static RG calculations [ 15,29,30] by calculating the two-loop dynamical RG function ζ Γ given in Eq. (15). Combining this result with the former data for the static critical behaviour we obtained numerical values for the effective critical exponent z eff which governs the critical slowing down of the relaxational time when T c is approached. In Figs. 3,4 we give results for two most physically interesting cases m = 2 and m = 3, which correspond to the easy-plane and Heisenberg random anisotropy magnets. Although the asymptotic dynamical critical behaviour of random anisotropy systems with cubic distribution is the same as for the random-site Ising systems, the crossover between different fixed points considerably influences the non-asymptotic critical properties. Different scenarios of dynamical critical behaviour are observed. Since the approach to asymptotics is very slow, it might be observed in real and numerical experiments. The effective exponents measured may take values essentially differing form the asymptotic one (in our calculation z = 2.139). For example in a large region z eff can be equal to the exponent of the pure Ising model (z = 2.052). One more particular feature of critical slowing down in random anisotropy magnets which is predicted by our analysis is that, contrary to the diluted magnets, z eff seems to reach its asymptotic value z always from the region z eff > z.
Another important contribution to the effective dynamical exponent could come from the coupling of the order parameter to a conserved density (changing from model A dynamics to model C [ 4]). Since the stable fixed point has a non-diverging specific heat the asymptotic discussed here would not be changed [ 39]. A more detailed account on that is in preparation.
It is our pleasure and honour to contribute this paper to the conference on the occasion of Prof. I.R. Yukhnovskii 80th birthday. His early work on the phase transitions theory and on an account of the non-asymptotic criticality [ 5] preceded in many respects many later contributions to this field. One of us (Yu.H.) is deeply indebted to the jubilee for introducing into the fascinating field of phase transitions and critical phenomena. R. F. acknowledges the fruitful cooperation with the Institute for Condensed Matter Physics. This work was supported by Austrian Fonds zur Förderung der wissenschaftlichen Forschung under Project No. P16574.