Model C critical dynamics of random anisotropy magnets
Model C critical dynamics of random anisotropy
magnets
M. Dudka1,2, R. Folk2, Yu. Holovatch1,2 and G. Moser3
1 Institute for Condensed Matter Physics, National Acad. Sci. of Ukraine, UA-79011
Lviv, Ukraine
2 Institut für Theoretische Physik, Johannes Kepler Universität Linz, A–4040 Linz,
Austria
3 Institut für Physik und Biophysik, Universität Salzburg, A–5020 Salzburg, Austria
Abstract. We study the relaxational critical dynamics of the three-dimensional
random anisotropy magnets with the non-conserved n-component order parameter
coupled to a conserved scalar density. In the random anisotropy magnets the
structural disorder is present in a form of local quenched anisotropy axes of random
orientation. When the anisotropy axes are randomly distributed along the edges
of the n-dimensional hypercube, asymptotical dynamical critical properties coincide
with those of the random-site Ising model. However structural disorder gives rise
to considerable effects for non-asymptotic critical dynamics. We investigate this
phenomenon by a field-theoretical renormalization group analysis in the two-loop order.
We study critical slowing down and obtain quantitative estimates for the effective and
asymptotic critical exponents of the order parameter and scalar density. The results
predict complex scenarios for the effective critical exponent approaching an asymptotic
regime.
PACS numbers: 05.50.+q, 05.70.Jk, 61.43.-j, 64.60.Ak, 64.60.Ht
E-mail: maxdudka@icmp.lviv.ua, hol@icmp.lviv.ua, folk@tphys.uni-linz.ac.at
http://arxiv.org/abs/0704.0896v1
Model C critical dynamics of random anisotropy magnets 2
1. Introduction
In this paper, we address the peculiarities of criticality under an influence of the random
anisotropy of structure. To be more specific, given a reference system is a 3d magnet with
n-component order parameter which below the second order phase transition point Tc
characterizes a ferromagnetic state, what will be the impact of random anisotropy [1–3]
on the critical dynamics [4,5] of this transition? It appears, that contrary to the general
believe that even weak random anisotropy destroys ferromagnetic long-range order at
d = 3, this is true only for the isotropic random axis distribution [6]. Therefore, we will
study a particular case, when the second order phase transition survives and, moreover,
it remains in the random-Ising universality class [7, 8] for any n. A particular feature
of 3d systems which belong to the random-Ising universality class is that their heat
capacity does not diverge at Tc (it is the isothermal magnetic susceptibility which
manifests singularity) [9]. Again, general arguments state [10,11] that for such systems
the relaxational critical dynamics of the non-conserved order parameter coupled to a
conserved density, model C dynamics, degenerates to purely relaxation model without
any couplings to conserved densities (model A). Nevertheless, this statement is true only
in the asymptotics [12,13] (i.e. at Tc, which in fact is never reached in experiments or in
simulations). As we will show in the paper, common influence of two different factors:
randomness of structure and coupling of dynamical modes leads to a rich effective critical
behavior which possesses many new unexpected features.
Dynamical properties of a system near the critical point are determined by the
behavior of its slow densities. In addition to the order parameter density ϕ these are
the conserved densities. Here, we consider the case of one conserved density m. For the
description of critical dynamics the characteristic time scales for the order parameter,
tϕ, and for the conserved density, tm, are introduced. Approaching the critical point,
where the correlation length ξ is infinite, they are growing accordingly to the scaling
tϕ ∼ ξz, (1)
tm ∼ ξzm . (2)
These power laws define the dynamical critical exponents of the order parameter, z,
and of the conserved densities, zm. The conserved density dynamical exponents may be
different from that of the order parameter.
The simplest dynamical model taking into account conserved densities is model
C, [4, 14] which contains a static coupling between non-conserved n-dimensional order
parameter ϕ and scalar conserved density m. Being quite simple, the model can be
applied to the description of different physical systems. In particular, in a lattice
model of intermetallic alloys [15] the non-conserved order parameter corresponds to
differences in the concentration of atoms of certain kind between the odd and even
sublattices. It is coupled to a conserved quantity – the concentration of atoms of
this kind in the full system. In the supercooled liquids the fraction of locally favored
Model C critical dynamics of random anisotropy magnets 3
structures is non-conserved “bond order parameter”, coupled to the conserved density
of a liquid [17]. Systems containing annealed impurities with long relaxational times [18]
manifest certain similarity with the model C as well.
Dynamical properties of a model with coupling to a conserved density were less
studied numerically than those for model without any coupling to secondary densities.
It may be the consequence of the complexity of the numerical algorithms, which turn
out to be much slower than for the simpler model. Simulations were performed for an
Ising antiferromagnet with conserved full magnetization and non-conserved staggered
magnetization (i.e. the order parameter) [19] and also for an Ising magnet with conserved
energy [20].
Theoretical analysis of model C critical dynamics were performed by means of
the field-theoretical renormalization group. Critical dynamical behavior of model C in
different regions of d − n plane was analyzed by ε = 4 − d expansion in first order in
ε [14]. The results lead to speculations about the existence of an anomalous region
for 2 < n < 4, where the order parameter is much faster than the conserved density
and dynamic scaling is questionable. Recent two-loop calculation [21, 22] corrected the
results of Ref. [23] and showed an absence of the anomalous region 2 < n < 4.
For the 3d model C with order parameter dimension n = 1, the conserved density
lead to the ”strong” scaling: [21,22] the dynamical exponents z and zm coincide and are
equal 2 + α/ν, where α and ν are the specific heat and the correlation length critical
exponents, correspondingly. For the Ising system (n = 1) the specific heat diverges
and α > 0. While for a system with α < 0, that is for the physically interesting
cases n = 2, 3, the scalar density decouples from the order parameter density in the
asymptotic region. It means that for such values of n the order parameter scales with
the same dynamical critical exponent z as in the model A and the dynamical exponent
of the scalar density is equal to zm = 2. The importance of the sign of α was already
mentioned in Ref. [14].
A rich critical dynamical behavior has already been observed in system with
structural disorder [18, 24–27]. Interest in this case is increased by the fact that real
materials are always characterized by some imperfection of their structure. Obviously,
that models describing their properties should contain terms connected with structural
disorder of certain type. For the static behavior of a system with quenched energy
coupled disorder (e.g. dilution), the Harris criterion [28] states that disorder does not
lead to a new static universality class if the heat capacity of the pure system does not
diverge, that is α < 0. In appears that in diluted systems α < 0 is always the case
(see Ref. [9]). The conclusion about influence of coupling between order parameter and
secondary density works also in this case. The presence of a secondary density does
not affect the dynamical critical properties in the asymptotics [10]: order parameter
dynamics is the same as in an appropriate model A, and zm = 2. Nevertheless, as we
noted at the beginning, the coupling between the order parameter and the secondary
density considerably influences the non-asymptotic critical behavior [12, 13].
We are interested in the critical dynamics of a systems with structural disorder of
Model C critical dynamics of random anisotropy magnets 4
another type, namely, random anisotropy magnets. Their properties are described by
the random anisotropy model (RAM) introduced in Ref. [1]. In this spin lattice model
each spin is subjected to a local anisotropy of random orientation, which essentially is
described by a vector and therefore is defined only for n > 1. The Hamiltonian reads: [1]
H = −
JR,R′ ~SR~SR′ − D̄
(x̂R~SR)
2, (3)
where, ~S = (S1, ..., Sn), are n-component vectors located on the sites R of a d-
dimensional cubic lattice, D̄ > 0 is an anisotropy constant, x̂ is a random unit vector
pointing in direction of the local anisotropy axis. The short-range interaction JR,R′ is
assumed to be ferromagnetic.
The static critical behavior of RAM was analyzed by many theoretical and
numerical investigations which could be compared with the critical properties of random
anisotropy magnets found in experiments (for recent review see Ref. [3]). The results
of this analysis bring about that random anisotropy magnets do not show a second
order phase transition for an isotropic random axis distribution. However they possibly
undergo a second-order phase transition for an anisotropic distribution (for references
see reviews Refs. [2,3]). Renormalization group studies of the asymptotic [7,8,29–31] and
non-asymptotic properties [3] of RAM corroborated such a conclusion. For example, the
RAM with random axes distributed due to the so-called cubic distribution was shown
within two-loop approximation to undergo a second order phase transition governed by
the random Ising critical exponents, [3, 8] as first suggested in Ref. [7]. Recently this
result found its confirmation in a five-loop RG study [31]. The cubic distribution allows
x̂ to point only along one of the 2n directions of the axes k̂i of a (hyper)cubic lattice: [29]
p(x̂) =
δ(n)(x̂− k̂i) + δ(n)(x̂+ k̂i)
, (4)
where δ(y) are Kronecker’s deltas.
Contrary to the static critical behavior of random anisotropy magnets their
dynamics was less investigated. Only dynamical models for systems with isotropic
distribution were briefly discussed in Refs. [32,33]. The critical dynamics was discussed
within model A, Ref. [34], and the dynamical exponents were calculated. However, it
does not give a comprehensive quantitative description since it is (i) restricted to the
isotropic distribution of the random axis and (ii) it is performed only within the first
non-trivial order of ε = 4− d expansion.
The model A critical dynamics of RAM with cubic random axis distribution was
analyzed within two-loop approximation in Ref. [35] Although the asymptotic dynamical
properties found coincide with those of the random-site Ising model, the non-asymptotic
behavior is strongly influenced by the presence of random anisotropy [35].
Beside the slow order parameter an additional slow conserved densities might be
present, for instance the energy density. Therefore considering the non-asymptotic
dynamical behavior of the RAM an extension to model C is of interest. Indeed, there
Model C critical dynamics of random anisotropy magnets 5
exist magnets where the distribution of the local random axes is anisotropic (e.g. the
rare earth compounds, see Ref. [3]).
The structure of the paper is as follows: Section 2 presents the equations defining
the dynamical model and its Lagrangian, the renormalization is performed is Section 3,
there the asymptotic and effective dynamical critical exponents are defined. In Section
4 we give the expressions for the field-theoretic functions in two-loop order and the
resulting non-asymptotic behavior is discussed. Section 5 summarizes our study. Details
of the perturbation expansion are presented in the appendix.
2. Model equations
Here we consider the dynamical model for random anisotropy systems described by
(3) with random axis distribution (4). The structure of the equations of motion for
n-component order parameter ~ϕ0 and secondary density [4, 14] m0 is not changed by
presence of random anisotropy
∂ϕi,0
= − Γ̊ ∂H
∂ϕi,0
+ θϕi , i = 1 . . . n, (5)
= λ̊∇2 ∂H
+ θm . (6)
The order parameter relaxes and conserved density diffuses with the kinetic coefficients
Γ̊, λ̊ correspondingly. The stochastic forces θϕi, θm obey the Einstein relations:
<θϕi(x, t)θϕj (x
′, t′)>= 2Γ̊δ(x− x′)δ(t− t′)δij , (7)
<θm(x, t)θm(x
′, t′)> =−2̊λ∇2δ(x−x′)δ(t−t′)δij . (8)
The disorder-dependent equilibrium effective static functional H describing behavior of
system in the equilibrium reads:
|∇~ϕ0|2+̊r̃|~ϕ0|2
|~ϕ0|4−D0(x̂~ϕ0)2 +
m20 +
γ̊m0|~ϕ0|2 − h̊m0
, (9)
where D0 is an anisotropy constant proportional to D̄ of Eq. (3), ˚̃r and ˚̃v depend on D̄
and the coupling of the usual φ4 model.
Integrating out the secondary density one reduces (9) to usual Ginzburg-Landau-
Wilson model with random anisotropy term and new parameter v̊ and r̊ connected to
the model parameters ˚̃r,˚̃v, γ̊ and h via relations:
r̊ =˚̃r + γ̊h̊, v̊ =˚̃v − 3̊γ2 (10)
We study the critical dynamics by applying the Bausch-Janssen-Wagner approach
[36] of dynamical field-theoretical renormalization group (RG). In this approach, the
critical behavior is studied on the basis of long-distance and long-time properties of
the Lagrangian incorporating features of dynamical equations of the model. The model
defined by expressions (5)-(9) within Bausch-Janssen-Wagner formulation [36] turns out
Model C critical dynamics of random anisotropy magnets 6
to be described by an unrenormalized Lagrangian:
ϕ̃0,iϕ̃0,i+
ϕ̃0,i
+Γ̊(̊µ̃−∇2)
ϕ0,i +
λ̊m̃0∇2m̃0 + m̃0
− λ̊∇2
Γ̊̊ṽϕ̃0,iϕ0,i
ϕ0,jϕ0,j +
2Γ̊D0(x̂ϕ̃0,i(t))(x̂ϕ0,i(t)) + Γ̊γ̊m0ϕ̃0,iϕ0,i−
λ̊̊γm̃0∇2ϕ0,iϕ0,i
,(11)
with auxiliary response fields ϕ̃i(t). There are two ways to average over the disorder
configurations for dynamics. The first way originates from statics and consists in using
the replica trick, [37] where N replicas of the system are introduced in order to facilitate
configurational averaging of the corresponding generating functional. Finally the limit
N → 0 has to be taken.
However we follow the second way proposed in Ref. [33]. There it was shown that
the replica trick is not necessary if one takes just the average of the Lagrangian with
respect to the distribution of random variables. The Lagrangian obtained in this way is
described by the following expression:
ϕ̃i,0
+Γ̊(̊µ̃−∇2)
ϕi,0−Γ̊ϕ̃i+
Γ̊̊ṽ
ϕj,0ϕj,0+
ϕ3i,0
+λ̊m̃0∇2m̃0 + m̃0
− λ̊∇2
m0 + Γ̊γ̊m0ϕ̃i,0ϕi,0−
λ̊γ̊m̃0∇2ϕi,0ϕi,0+
ϕ̃i,0(t)ϕi,0(t)
Γ̊2ů
ϕ̃j,0(t
′)ϕj,0(t
Γ̊2ẘ
ϕ̃i,0(t
′)ϕi,0(t
. (12)
In Eq. (12), the bare mass is ˚̃µ = ˚̃r − D/n, and bare couplings are ů > 0, ˚̃v > 0,
ẘ < 0. Terms with couplings ů and ẘ are generated by averaging over configurations
and the values of ů and ẘ are connected to the moments of distribution (4). Therefore
the ratio of the two couplings has to be ẘ/ů = −n. The ẙ-term in (12) does not
result from the averaging procedure but has to be included since it is generated in the
perturbational treatment. It can be of either sign.
3. RG functions
We perform renormalization within minimal subtraction scheme introducing renormal-
ization factors Zai , ai = {{α}, {δ}}, leading to the renormalized parameters {α} =
{u, v, w, y, γ,Γ, λ} and renormalized densities {δ} = {ϕ, ϕ̃,m, m̃}. For the specific heat
we need also an additive renormalization Aϕ2 which leads to the function
Bϕ2(u,∆) = µ
εZ2ϕ2µ
µ−εAϕ2
, (13)
Model C critical dynamics of random anisotropy magnets 7
with the scale parameter µ and factor Zϕ2 that renormalizes the vertex with ϕ
2 insertion.
From the Z-factors one obtains the ζ-functions describing the critical properties
ζa({α}) = −
d lnZa
d lnµ
, (14)
Relations between the renormalization factors lead to corresponding relations between
the ζ-functions. In consequence for the description of the critical dynamics one needs
only ζ-functions of the couplings, ζui (ui = {u, v, w, y} for i = 1, 2, 3, 4), the order
parameter ζϕ, the auxiliary field ζϕ̃, ϕ
2-insertion ζϕ2 and also function Bϕ2 . In particular,
the ζ-function of the time scale ratio
introduced for the description of dynamic properties is related to the above ζ-functions:
ζϕ̃ − γ2Bϕ2 . (16)
The behavior of the model parameters under renormalization is described by the
flow equations
= β{α} . (17)
The β-functions for the static model parameters have the following explicit form:
βui = ui(ε+ ζϕ + ζui), (18)
βγ = γ(
+ ζϕ2 +
Bϕ2). (19)
The dynamic β-function for the time scale ratio W reads
βW = WζW = W (
ζϕ̃ − γ2Bϕ2). (20)
The asymptotic critical behavior of the system is obtained from the knowledge of
the fixed points (FPs) of the flow equations (17). A FP {α∗} = {u∗, v∗, w∗, y∗, γ∗,W ∗}
is defined as simultaneous zero of the β-functions. The set of equations for the static
fourth order couplings decouple from the other β-functions. Thus for each of the FPs
of the static forth order couplings {u∗i } one obtains two FP values of the static coupling
between the order parameter and the conserved density γ:
=0 and γ∗
ε− 2ζϕ2({u⋆i })
Bϕ2({u⋆i })
νBϕ2({u⋆i })
, (21)
where α and ν are the heat capacity and correlation length critical exponent calculated
at the corresponding FP {u∗}. Inserting the obtained values for the static FPs into the
β-function (20) one finds the corresponding FP values of the time scale ratio W .
The stable FP accessible from the initial conditions corresponds to the critical
point of system. A FP is stable if all eigenvalues ωi of the stability matrix ∂βαi/∂αj
calculated at this FP have positive real parts. The values of ωi indicate also how fast
the renormalized model parameters reach their fixed point values.
Model C critical dynamics of random anisotropy magnets 8
From the structure of β-functions we conclude, that the stability of any FP with
respect to the parameters γ and W is determined solely by the derivatives of the
corresponding β-functions:
, ωW =
. (22)
Moreover using (19) we can write:
ωγ = −
ε− 2ζϕ2({ui})
γ2Bϕ2(u,∆) , (23)
which at the FP {α∗} leads to:
ωγ|{α}={α∗} = −
for γ∗
= 0 , (24)
ωγ|{α}={α∗} =
for γ∗
2 6= 0 . (25)
Therefore, a stability with respect to parameter γ is determined by the sign of the
specific heat exponent α. For a system with non-diverging heat capacity (α < 0) at the
critical point, γ∗ = 0 is the stable FP. Static results report that the stable and accessible
FP is of a random site Ising type. In this case α < 0. This leads to the conclusions that
in the asymptotic region the secondary density decouples from the order parameter.
The critical exponents are defined by the FP values of the ζ-functions. For instance,
the asymptotic dynamical critical exponent z is expressed at the stable FP by:
z = 2 + ζΓ({α∗}), (26)
ζΓ({α}) =
ζϕ({ui})−
ζϕ̃({α}). (27)
In similar way the dynamical critical exponent zm for the secondary density is defined
zm = 2 + ζm({u∗i }, γ∗), (28)
where
ζm({ui}, γ) =
γ2Bϕ2({ui}). (29)
While their effective counterparts in the non asymptotic region are defined by the
solution of flow equations (17) as
zeff = 2 + ζΓ({ui(ℓ)}, γ(ℓ),W (ℓ)), (30)
zeffm = 2 + ζm({ui(ℓ)}, γ2(ℓ)). (31)
In the limit ℓ → 0 the effective exponents reach their asymptotic values. In the next
section we analyze the possible scenarios of effective dynamical behavior as well as check
the approach to the asymptotical regime.
Model C critical dynamics of random anisotropy magnets 9
4. Results
4.1. Asymptotic properties
The static two-loop RG functions of RAM with cubic random axis distribution in the
minimal substraction scheme agree with the results obtained in Ref. [3] using the replica
trick and read:
βu= − εu+
n + 2
uw+yu+
11 (n+ 2)
−5 (n + 2)
v2u−11
u2w− 5
uw2−11
u2y − 5
vuw −
vuy −
v2w −
w2v, (32)
βv= − εv+
v2+2vu+
wv+yv−
3n+ 14
11n+58
wvu, (33)
βw= − εw+
w2+2wu+
wv+yw−7
w3−29
w2u−41
wu2−31
v2w−11
w2y− 5
y2w−17
wuy−5n+ 34
wvu−5
vwy, (34)
βy = − εy +
y2 + 2yu+ 2yv + 2wy +
wv−17
y3 − 41
y2u−23
y2v−23
y2w−5n+ 82
v2y−41
w2y−2w2v−
n + 18
5n+ 82
vuw, (35)
, (36)
u2− 5
n + 2
12v2−5
wu−5yu
. (37)
Here, u, v, w, y stand for the renormalized couplings.
Given the expression for the function ζϕ2 , Eq. (37), the function βγ can be
constructed via Eq. (19) and the two-loop expression Bϕ2 = n/2.
In order to discuss the dynamical FPs it turns out to be useful to introduce the
parameter ρ = W/(1 + W ) which maps W and its FPs into a finite region of the
Model C critical dynamics of random anisotropy magnets 10
parameter space ρ. Then instead of the flow equation for W the flow equation for ρ
arises in (17):
= βρ({ui}, γ, ρ), (38)
where according to (20)
βρ({ui}, γ, ρ) = ρ(ρ− 1)(ζΓ({ui}, γ, ρ)− γ2Bϕ2({ui})). (39)
The function ζΓ in the above expression is obtained from Eq (27) using the static
function ζϕ (36) and the two loop result for the dynamic function ζϕ̃ (calculated from
Eq (A.8)). We get the following two-loop expression for ζΓ:
ζΓ = −
+ γ2ρ+
(6 ln (4/3)− 1)
(n + 2)
vy + y2
5u2 + (n + 2)uv + 10uw + 3uy + 3vw + 5w2 + 3wy
(n+ 2)
v + y
(1− 3 ln (4/3)) +
3 (n+ 2)
ln (4/3) + (1 + ρ) ln
1− ρ2
+ u+ w
ρ2 ln (ρ)
+ (3 + ρ) ln (1− ρ)
. (40)
The two-loop result [22] for the pure model C is recovered by setting in (40) the couplings
u, w, y equal to zero. While setting γ = 0 in (40) the result for model A with random
anisotropy [35] is recovered. The γ2u, γ2w, γ2y-terms represent the intrinsic contribution
of model C for random anisotropy magnets.
There are two different ways to proceed with the numerical analysis of the
perturbative expansions for the RG functions (32) - (35), (40). The first one is an
ε-expansion [38] whereas the second one is the so-called fixed-dimension approach [39].
In the frames of the latter approach, one fixed ε and solves the non-linear FP equations
directly at the space dimension of interest (i.e. at ε = 1 in our d = 3 case). Whilst
in many problems these two ways serve as complementing ones, it appears that for
certain cases only one of them, namely the fixed-d approach leads to the quantitative
description. Indeed, as it is well known by now, the ε-expansion turns into the
expansion for the random-site Ising model and no reliable numerical estimates can be
obtained on its basis (see [9] and references therein). As one will see below, the random-
site Ising model behavior emerges in our problem as well, therefore we proceed within
the fixed-d approach.
The series for RG functions are known to diverge. Therefore to obtain reliable
results on their basis we apply the Padé-Borel resummation procedure [40] to the static
functions. It is performed in following way: we construct the Borel image of the resolvent
series [41] of the initial RG function f :
0≤i+j+l+k≤2
ai,j,k,l(ut)
i(vt)j(wt)k(yt)l →
Model C critical dynamics of random anisotropy magnets 11
0≤i+j+l+k≤2
ai,j,k,lu
ivjwkylti+j+k+l
Γ(i+ j + 1)
where f stands for one of the static RG functions βui , βγ/γ − γ2n/4, ai,j,k,l are the
corresponding expansion coefficients given by Eqs. (32)–(37), and Γ(i+ j+1) is Euler’s
gamma function. Then, the Borel image is extrapolated by a rational Padé approximant
[42] [K/L](t). Within two-loop approximation we use the diagonal approximant with
linear denominator [1/1]. As it is known in the Padé analysis, the diagonal approximants
ensure the best convergence of the results [42]. The resummed function is then calculated
by an inverse Borel transform of this approximant:
f res =
dt exp(−t)[1/1](t). (41)
As far as the above procedure enables one to restore correct static RG flow (as sketched
below) we do not further resum the dynamic RG function βW .
The analysis of the static functions β{ui} at fixed dimension d = 3 brings about an
existence of 16 FPs [3, 31]. Only ten of these FPs are situated in the region of physical
interest u > 0, v > 0, w < 0. Corresponding values of FP coordinates can be found in
Ref. [3].
For each static FP {u∗i} we obtain a set of dynamical FPs with different γ∗ and
ρ∗. The FPs obtained for n = 2, 3 are listed in Table 1 and Table 2 correspondingly.
Stability exponents ωγ and ωρ are given in tables as well. Here we keep the numbering
of FPs already used in Refs. [3, 8, 29, 31, 35]. It is known from the statics that FP
XV governs the critical behavior of RAM with cubic distribution. This FP is of the
same origin as the FP of random-site Ising model therefore all static critical exponents
coincide with those of the random-site Ising model. Since the specific heat exponent in
this case is negative, the asymptotic critical dynamics is described by model A. However
the non-asymptotic critical properties of random anisotropy magnets are different from
the random-site Ising magnets in statics [3] as well as in dynamics [35]. Moreover, for
the model C considered here, the non-asymptotic critical behavior differs considerably
from that of the corresponding model A as we will se below.
4.2. Non-asymptotic properties
The existence of such a large number of dynamical FPs makes non-asymptotic critical
behavior more complex as in model A. We present here results for n = 3. For n = 2
the behavior is qualitatively similar. Solving the flow equations for different initial
conditions we obtain different flows in the space of model parameters. The projection
of most characteristic flows into the subspace w−y−ρ is presented in Fig. 1. The open
circles indicate genuine model C unstable FPs whereas filled circles represent model A
unstable FPs. The filled square denotes the stable FP.
The initial conditions for the couplings u(0), v(0), w(0), y(0) for the flows shown
are the same as those in Refs. [3, 35]. We choose γ(0) = 0.1 and ρ(0) = 0.6. Many
flows are affected by the two Ising FPs VC and XC . Inserting the solutions of the flow
Model C critical dynamics of random anisotropy magnets 12
-1,8-1,6
Figure 1. Projections of flows for n = 3 in the subspace of couplings w− y− ρ. Open
circles represent projections of unstable FPs with non-zero γ∗. Filled circles denote
unstable FPs with γ∗ = 0. The filled square shows the stable FP. See Section 4.2 for
a more detailed description.
-100 -80 -60 -40 -20 0
Figure 2. Dependence of the order parameter effective dynamical critical exponent in
the model C dynamics on the logarithm of flow parameter. See text for full description.
-100 -80 -60 -40 -20 0
Figure 3. Dependence of the conserved density effective dynamical critical exponent in
the model C dynamics on the logarithm of flow parameter. See text for full description.
Model C critical dynamics of random anisotropy magnets 13
Table 1. Two-loop values for the dynamical FPs of random anisotropy magnets with
n = 2 (model C).
FP γ∗ ρ∗ ωγ ωρ z
I 0 0 ≤ ρ∗ ≤ 1 0 0 2
I′ 1 0 1 -1 2
IC 1 0.6106 1 0.745 3
I′1 1 1 1 -∞ ∞
II 0 0 0.0387 0.0526 2.0526
II1 0 1 0.0387 -0.0526 2.0526
III 0 0 -0.1686 -0.1850 1.8150
III1 0 1 -0.1686 0.1850 1.8150
IIIC .5806 0 0.3371 -0.5222 1.8150
III′1 .5806 1 0.3371 ∞ −∞
V 0 0 -0.0525 0.0523 2.0523
V1 0 1 -0.0525 -0.0523 2.0523
V′ .3240 0 0.1050 -0.0527 2.0523
VC .3240 0.5241 0.1050 0.0277 2.1050
V′1 .3240 1 0.1050 -∞ ∞
VI 0 0 -0.0049 -0.0417 2.0107
VI1 0 1 -0.0049 0.0417 2.0107
VI′ 0.0986 0 0.0097 0.00095 2.0107
VI′1 0.0986 1 0.0097 ∞ -∞
VIII 0 0 -0.0525 0.1569 2.1569
VIII1 0 1 -0.0525 -0.1569 2.1569
VIII′ 0.3240 0 0.1050 0.0519 2.1569
VIII′1 0.3240 1 0.1050 -∞ ∞
X 0 0 -0.0525 0.0523 2.0523
X1 0 1 -0.0525 -0.0523 2.0523
X′ .3240 0 0.1050 -0.0527 2.0523
XC .3240 0.5241 0.1050 0.0277 2.1050
X′1 .3240 1 0.1050 -∞ ∞
XV 0 0 0.0018 0.1388 2.1388
XV1 0 1 0.0018 -0.1388 2.1388
equations into the expressions for dynamical exponents we obtain the effective exponents
zeff and zeffm . The dependence of z
eff on the flow parameter ℓ corresponding to flows 1-7
is shown in Fig. 2. Similarly Fig 3 shows this dependence for the effective exponent of
the conserved density zeffm . Flow 3 is affected by both FPs VC and XC . Therefore the
effective exponents demonstrate a region with values which are close to those for model
C in the case of the Ising magnet (see curves 3 in Figs. 2 and 3). The asymptotic values
corresponding to the FPs VC and XC are indicated by the dashed line. They correspond
Model C critical dynamics of random anisotropy magnets 14
Table 2. Two-loop values for the dynamical FPs of random anisotropy magnets with
n = 3 (model C).
FP γ∗ ρ∗ ωγ ωρ z
I 0 0 ≤ ρ∗ ≤ 1 0 0 2
I′ 0.8165 0 1 -1 2
IC 0.8165 0.7993 1 0.5218 3
I′1 0.8165 1 1 -∞ ∞
II 0 0 0.1109 0.0506 2.0506
II1 0 1 0.1109 -0.0506 2.0506
III 0 0 -0.1686 -0.1850 1.8150
III1 0 1 -0.1686 0.1850 1.8150
III′ 0.4741 0 .3371 -0.5222 1.8150
III′1 0.4741 1 0.3371 ∞ −∞
V 0 0 -0.0525 0.0523 2.0523
VI1 0 1 -0.0525 -0.0523 2.0523
VI′ 0.2646 0 0.1050 -0.0527 2.0523
VIC 0.2646 0.7617 0.1050 0.0157 2.1050
VI′1 0.2646 1 0.1050 -∞ ∞
VI 0 0 -0.0162 -0.0401 1.9599
VI1 0 1 -0.0162 0.0401 1.9599
VI′ 0.1467 0 0.0323 -0.0724 1.9599
VI′1 0.1467 1 0.0323 ∞ -∞
VIII 0 0 0.1051 0.0425 2.0425
VIII1 0 1 0.1051 -0.0425 2.0425
IX 0 0 -0.0161 -0.0384 1.9616
IX1 0 1 -0.0161 0.0384 1.9616
IX′ 0.1466 0 0.0322 -0.0707 1.9616
IX′1 0.1466 1 0.0322 ∞ -∞
X 0 0 -0.0525 0.0523 2.0523
X1 0 1 -0.0525 -0.0523 2.0523
X′ 0.2646 0 0.1050 -0.0527 2.0523
XC 0.2646 0.7617 0.1050 0.0157 2.1050
X′1 0.2646 1 0.1050 -∞ ∞
XV 0 0 0.0018 0.1388 2.1388
XV1 0 1 0.0018 -0.1388 2.1388
Model C critical dynamics of random anisotropy magnets 15
to the values asymptotically obtained in the pure model C with n = 1, since the FPs
VC and XC are of the same origin, that FP of pure model C. Curves 6 correspond to
flows near the pure FP II. Whereas curve 7 corresponds to the flow near the cubic FP
VIII.
The main difference of the behavior of the effective dynamical exponent zeff in
model C from that in model A is the appearance of curves with several peaks. The
value of the peak appearing on the right-hand side depends on the initial condition γ(0)
and ρ(0). This is demonstrated in Fig.4.
-100 -80 -60 -40 -20 0
 γ=0.01, ρ=0.1
 γ=0.01, ρ=0.6
 γ=0.1, ρ=0.1
 γ=0.1, ρ=0.6
 γ=0.6, ρ=0.1
 γ=0.6, ρ=0.6
Figure 4. Dependence of zeff on the logarithm of flow parameter for different initial
values γ and ρ
-100 -80 -60 -40 -20 0
Figure 5. Normalized effective dynamical critical exponents of order parameter and
conserved density zeff (solid line), zeffm (dashed line) correspondent to flow 2.
The effective behavior of the two dynamical critical exponents for the order
parameter and the conserved density might be quite different as one sees comparing
Figs. 2 and 3. However, one may ask if both exponents reach the asymptotic values
in the same way. For this purpose we introduce a normalization of the values of the
effective exponents by their values in the asymptotics. In particular, we introduce
notations zeff = zeff/z, zeffm = z
m /zm for order parameter exponent and conserved
density exponent correspondingly. Figs. 5, 6 show behavior of normalized exponents for
order parameter and conserved density for flows 2 and 4 correspondingly. It illustrates
that approach to the asymptotics for order parameter exponents and conserved density
Model C critical dynamics of random anisotropy magnets 16
-100 -80 -60 -40 -20 0
Figure 6. Dependencies of normalized effective dynamical critical exponents of the
order parameter and conserved density correspondent to flow 4. Notations as if Fig.
exponents occurs in different way for different flows, that means for different initial
conditions. For system with small degree of disorder (small u(0) and w(0), flow 4) the
approach of order parameter dynamical exponent to asymptotic regime is faster than
for the conserved density one, while for system with larger amount of disorder (flow 2)
approach of both quantities is almost simultaneous.
5. Conclusion
In this paper, we have studied model C dynamics of the random anisotropy magnets
with cubic distribution of local anisotropy axis. For this purpose two-loop dynamical
RG function ζΓ has been obtained. On the base of static results [3] the dependencies of
effective critical exponents of order parameter, zeff , and conserved density, zeffm , on the
flow parameter were calculated.
The two-loop approximation adopted in our paper may be considered as certain
compromise between what is feasible in static calculations from one side, and in dynamic
ones form the other side. As a matter of fact, the state-of-the-art expansions of the static
RG functions in the minimal subtraction scheme are currently available for many models
within the five-loop accuracy [9, 44] but it is not the case for the dynamic functions.
Complexity of dynamical calculations is reflected in the current situation, when the
results beyond two loops have been obtained for model A only. The model C even with
no structural disorder seems to be outside present manageable problems (see the recent
review [5]). However, there are examples which demonstrate the even in two loops
highly accurate results for dynamical characteristics can be obtained. One of them is
given by the critical dynamics of 4He at the superfluid phase transition [45]. Besides,
analysis of the two-loop static RG functions refined by resummation also brings about
sufficiently accurate quantitative characteristics of a static critical behavior in disordered
systems [9, 46].
In the asymptotics the conserved density is decoupled from the order parameter
and the dynamical critical behavior of random anisotropy model with cubic random axis
distribution is the same as that of the random-site Ising model. Crossover occurring
Model C critical dynamics of random anisotropy magnets 17
between different FPs present in the random anisotropy model considerably influences
the non-asymptotic critical properties. Different scenarios of dynamical critical behavior
are observed depending of the initial values of the model parameters. The main feature
is the presence of additional peaks on the curves for the effective dynamical critical
exponents in comparison with the effective model A critical dynamics.
As far as the approach to the asymptotics is very slow, the effective exponents may
be observed in experiments and in numerical simulations. The effective exponent for the
order parameter may take a value far away from the asymptotic one (the asymptotic
value in in our two loop calculation is z = 2.139). The same holds for the conserved
density effective critical exponent which may be far of its van Hove asymptotic value
zm = 2. For example one can observe values of z
eff and zeffm close to those for pure Ising
model with model C dynamics.
This work was supported by Fonds zur Förderung der wissenschaftlichen Forschung
under Project No. P16574
Appendix A. Perturbation expansion
We perform our calculations on the basis of the Lagrangian defined by (12) using
the Feynman graph technique. The propagators for this Lagrangian are shown in the
Fig. A1.
k,w k ’ ’,w
G(k, ω)δ(k+k′)δ(ω+ω′)δi,j
k,w k ’ ’,w
C(k, ω)δ(k+k′)δ(ω+ω′)δi,j
k,w k ’ ’,w
i j H(k, ω)δ(k+k′)δ(ω+ω′)δi,j
k,w k ’ ’,w
i j D(k, ω)δ(k+k′)δ(ω+ω′)δi,j
Figure A1. Propagators for constructing Feynman graphs. G(k, ω) and H(k, ω) are
response propagators while C(k, ω) and D(k, ω) are correlation propagators.
Response propagators G(k, ω) and H(k, ω) are equal to
G(k, ω) = 1/(−iω + Γ̊(̊µ̃+ k2)) and H(k, ω) = 1/(−iω + λ̊k2) , (A.1)
while the correlation propagators C(k, ω) and D(k, ω) are equal to
C(k, ω) = 2Γ̊/|−iω+Γ̊(̊µ̃+k2)|2 and D(k, ω) = 2̊λk2/|−iω+λ̊k2|2 .(A.2)
The vertices defined by Lagrangian are shown in Fig. A2.
We obtain an expression for the two-point vertex function Γ̊
ϕ̃ϕ by keeping the
diagrams up to two-loop order. The result of calculations can be expressed in form:
Γ̊ϕ̃ϕ(ξ, k, ω) = −iωΩ̊ϕ̃ϕ(ξ, k, ω) + Γ̊stϕϕ(ξ, k)̊Γ . (A.3)
Here we introduce the correlation length ξ(µ̊ = ˚̃µ+ γ̊h̊, ů, v̊, ẘ, ẙ), which is defined by
∂ ln Γ̊stϕϕ
. (A.4)
Model C critical dynamics of random anisotropy magnets 18
k ’ ’,w
k ’’ ’’,w
k ’’’ ’’’,w
ΓAδ(k + k′ + k′′ + k′′′)δ(ω + ω′ + ω′′ + ω′′′)
k ’ ’,w
k ’’ ’’,w
k ’’’ ’’’,w
Γ2Bδ(k + k′)δ(k′′ + k′′′)δ(ω + ω′)δ(ω′′ + ω′′′)
c k,w
k ’ ’,w
k ’’ ’’,w
Γ̊γ̊δ(k + k′ + k′′)δ(ω+ω′+ω′′)δi,j
k ’ ’,w
k ’’ ’’,w
λ̊̊γk2δ(k + k′ + k′′)δ(ω+ω′+ω′′)δi,j
Figure A2. Vertices for our model. In vertex a, A stands for
v0/3! (δi,jδl,m + δi,lδj,m + δi,mδj,l)/3 or y0/3! δi,jδj,lδl,m. In vertex b, B stands for
u0/3! δi,jδl,m or w0/3! δi,jδj,lδl,m. Vertices c and d originate from the coupling to the
conserved density.
The function Γ̊ϕϕ is the static two-loop vertex function of the disordered magnet. The
structure (A.3) of the dynamic vertex function of pure model C was obtained in Ref. [43]
up to two-loop order.
We can express two-loop dynamical function Ω̊ϕ̃ϕ in the following form:
Ω̊ϕ̃ϕ(ξ, k, ω) = 1 + Ω̊
ϕ̃ϕ(ξ, k, ω) + Ω̊
ϕ̃ϕ(ξ, k, ω), (A.5)
where the one loop contribution reads:
Ω̊1ϕ̃ϕ(ξ, k, ω) = −
ů+ ẘ
(−iω + Γ̊(ξ−2 + k′2))(ξ−2+k′2)
+ γΓ̊IC(ξ, k, ω),(A.6)
while the two-loop contribution is of the form:
Ω̊2ϕ̃ϕ(ξ, k, ω) = Γ̊(
v̊2 +
ϕ̃ϕ (ξ, k, ω)−
v̊ + ẙ)̊γ2C̊
ϕ̃ϕ (ξ, k, ω) + Γ̊γ̊
4S̊ϕ̃ϕ(ξ, k, ω) +
v̊ů+
(CD2)
ϕ̃ϕ (ξ, k, ω) + Γ̊(
(CD3)
ϕ̃ϕ (ξ, k, ω) + W̊
(CD4)
ϕ̃ϕ (ξ, k, ω)
ů+ ẘ
(CD5)
ϕ̃ϕ (ξ, k, ω) + W̊
(CD6)
ϕ̃ϕ (ξ, k, ω) + 2W̊
(CD7)
ϕ̃ϕ (ξ, k, ω)
.(A.7)
In (A.6) and (A.7) the expressions for the integrals IC , W̊
(A), C̊(T3) and S̊ of the
pure model C are given in the Appendix A.1 in Ref. [22], while the contributions for
W̊ (CDi) are presented in the Appendix of Ref. [13].
Following the renormalization procedure for Γ̊ϕ̃ϕ we obtain the two-loop
renormalizating factor Zϕ̃:
Zϕ̃ = 1+
Model C critical dynamics of random anisotropy magnets 19
v + y
n + 2
v + y
1−3 ln
3(n+2)
(1+W )2
3(n+ 2)
vw−(u+ w)γ2 W
ln(1 +W )− 1
. (A.8)
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	Introduction
	Model equations 
	RG functions 
	Results 
	Asymptotic properties
	Non-asymptotic properties
	Conclusion
	 Perturbation expansion
