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...
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irk-123456789-1210482017-06-14T03:04:53Z Critical slowing down in random anisotropy magnets Dudka, M. Folk, R. Holovatch, Yu. Moser, G. 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 effects 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 zeff . Ми вивчаємо релаксаційну динаміку з незбережним параметром порядку (критична динаміка моделі А) для тривимірного магнетика з безладом у формі випадкової осі анізотропії. Для анізотропного розподілу випадкових осей асимптотична критична поведінка співпадає з поведінкою ізингівських систем з випадковими вузлами. Таким чином асимптотична критична динаміка керується динамічним показником випадкової моделі Ізинга. Однак безлад значно впливає на динамічну поведінку в неасимптотичному режимі. Ми проводимо теоретико-польовий ренормалізаційно-груповий аналіз в рамках схеми мінімального віднімання в двопетлевому наближенні щоб дослідити асимптотичну і ефективну критичну динаміку систем з випадковою анізотропією. Результати демонструють немонотонну поведінку динамічного ефективного критичного показника zeff . 2005 Article Critical slowing down in random anisotropy magnets / M. Dudka, R. Folk, Yu. Holovatch, G. Moser // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 737–748. — Бібліогр.: 39 назв. — англ. 1607-324X PACS: 05.50.+q, 05.70.Jk, 61.43.-j, 64.60.Ak, 64.60.Ht DOI:10.5488/CMP.8.4.737 http://dspace.nbuv.gov.ua/handle/123456789/121048 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
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 effects 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 zeff . |
| format |
Article |
| author |
Dudka, M. Folk, R. Holovatch, Yu. Moser, G. |
| spellingShingle |
Dudka, M. Folk, R. Holovatch, Yu. Moser, G. Critical slowing down in random anisotropy magnets Condensed Matter Physics |
| author_facet |
Dudka, M. Folk, R. Holovatch, Yu. Moser, G. |
| author_sort |
Dudka, M. |
| title |
Critical slowing down in random anisotropy magnets |
| title_short |
Critical slowing down in random anisotropy magnets |
| title_full |
Critical slowing down in random anisotropy magnets |
| title_fullStr |
Critical slowing down in random anisotropy magnets |
| title_full_unstemmed |
Critical slowing down in random anisotropy magnets |
| title_sort |
critical slowing down in random anisotropy magnets |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121048 |
| citation_txt |
Critical slowing down in random anisotropy magnets / M. Dudka, R. Folk, Yu. Holovatch, G. Moser // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 737–748. — Бібліогр.: 39 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT dudkam criticalslowingdowninrandomanisotropymagnets AT folkr criticalslowingdowninrandomanisotropymagnets AT holovatchyu criticalslowingdowninrandomanisotropymagnets AT moserg criticalslowingdowninrandomanisotropymagnets |
| first_indexed |
2025-07-08T19:06:03Z |
| last_indexed |
2025-07-08T19:06:03Z |
| _version_ |
1837106793245835264 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 737–748
Critical slowing down in random
anisotropy magnets
M.Dudka 1,2 , R.Folk 2 , Yu.Holovatch 1,2,3 , G.Moser 4
1 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
79011 Lviv, Ukraine
2 Institut für Theoretische Physik,
Johannes Kepler Universität Linz,
A–4040, Linz, Austria
3 Ivan Franko National University of Lviv,
79005 Lviv, Ukraine
4 Institut für Physik und Biophysik,
Universität Salzburg,
A–5020 Salzburg, Austria
Received July 18, 2005
We study the purely relaxational critical dynamics with non-conserved or-
der parameter (model A critical dynamics) for three-dimensional magnets
with disorder in a form of the random anisotropy axis. For the random ax-
is anisotropic distribution, the static asymptotic critical behaviour coincides
with that of random site Ising systems. Therefore the asymptotic critical dy-
namics is governed by the dynamical exponent of the random Ising mod-
el. However, the disorder effects 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 approxi-
mation to investigate asymptotic and effective critical dynamics of random
anisotropy systems. The results demonstrate the non-monotonic behaviour
of the dynamical effective critical exponent zeff .
Key words: critical dynamics, disordered systems, random anisotropy,
renormalization group
PACS: 05.50.+q, 05.70.Jk, 61.43.-j, 64.60.Ak, 64.60.Ht
1. 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 experimen-
tal 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
c© M.Dudka, R.Folk, Yu.Holovatch, G.Moser 737
M.Dudka et al.
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 which also
causes the relaxation time τ to diverge with the dynamical critical exponent z:
τ ∼ ξz. (1)
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].
However, this holds 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 in observing the behaviour in certain universality class-
es is the homogeneity of the system under consideration. Therefore the effect of im-
purities on critical behaviour is of considerable interest. However it turned out that
the disordered systems may also show scaling within a certain universality class.
This might be the universality class of a pure system or a new one [6–9]. Moreover
the changes introduced by the disorder depend on the type of this disorder; name-
ly, whether they are 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 noncorrelated. It may even happen that
the second order transition of the pure system is destroyed [16,17].
It 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 be-
haviour 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) the
effective critical behaviour is capable of explaining the experimental situation [11,18].
RG investigations of dynamic critical behaviour is in many cases technically
much more involved in comparison with statics. Thus, the results for dynamics are
known 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 the 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 alloys
are recognized as random anisotropy magnets [14], certain crystalline compounds
with rear-earth component belong to this class as well [19]. Random anisotropy also
characterizes the molecular based magnets [20], nanocrystalline materials [21], as
well as granular systems [22]. Moreover, the analysis involving random anisotropy
also found its application in interpreting 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 being
738
Critical slowing down in random anisotropy magnets
subjected to a local anisotropy of random orientation. The Hamiltonian of the ran-
dom anisotropy model (RAM) reads:
H = −
∑
R,R′
JR,R′
~SR
~SR′ − D0
∑
R
(x̂R
~SR)2. (2)
Here, ~SR = (S1
R
, . . . , Sm
R
), vectors R span sites of a d–dimensional cubic lattice,
D0 > 0 is an anisotropy constant, x̂ is a random unit vector specifying the direction
of the local anisotropy axis. The interaction JR,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 vectors x̂R in (2) are ran-
domly distributed with a distribution function p(x̂) and fixed in a certain config-
uration. It is well established by now, that the anisotropy axis distribution plays
a crucial role in originating the low-temperature phase in the RAM. In particular,
when the random vectors x̂R point with equal probability towards any direction,
such a distribution may be called an isotropic one, the ferromagnetic ordering is im-
possible [25] for spatial dimension d 6 4. However, it may occur for an anisotropic
distribution. In statics, this situation was corroborated by the RG studies of RAM
[26–30] restricting x̂ to be pointed along one of the 2m directions of the axes k̂i of
a hypercubic lattice (cubic distribution):
p(x̂) =
1
2m
m
∑
i=1
[
δ(m)(x̂ − k̂i) + δ(m)(x̂ + k̂i)
]
(3)
with Kronecker deltas δ(y). In this case, there occurs the second order phase tran-
sition into the magnetically ordered low-temperature phase. Asymptotically it is
characterized by the critical exponents of the random-site Ising model as suggested
already in [27] and confirmed later in [28–30].
The studies of static criticality of random anisotropy magnets are far from being
as intensive as those of the diluted magnets [10], and even less is known about
their dynamic critical behaviour. The dynamical models for systems with isotrop-
ic distribution of local anisotropy axis were considered in [31–33], the first order
RG calculations were performed in [34]. However, the problem of dynamic critical
behaviour of a RAM with an anisotropic random axis distribution has remained
untouched so far.
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
[4]) and a cubic random axis distribution (3). The static critical behaviour of such
magnets (note, with m > 1) belongs to the universality class of a random-site
Ising model [27–30], for which the heat capacity does not diverge [10]. Therefore,
the critical dynamics of such a model for any m is governed in asymptotics by
the model A random-site Ising magnet dynamical critical exponent. However in the
non-asymptotic region, the model possesses a rich effective critical behaviour, as will
be shown by our subsequent analysis. Assuming that it is this effective behaviour
739
M.Dudka et al.
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 follows: in the next section 2 we present the
Langevin equations governing the model A dynamics and describe the renormaliza-
tion procedure. In section 3 we give results of our calculations obtained in two-loop
approximation and display possible scenarios of effective critical behaviour. Conclu-
sions and outlook are given in section 4.
2. 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 ~ϕ0 ≡ ~ϕ0(R). In this
case the relaxation of the order parameter is described by the Langevin equation:
∂ϕi,0
∂t
= −Γ̊
∂H
∂ϕi,0
+ θϕi
i = 1, . . . ,m, (4)
with the Onsager coefficient Γ̊, stochastic forces θϕi
obeying the Einstein relations:
〈
θϕi
(R, t)θϕj
(R′, t′)
〉
= 2Γ̊δ(R − R′)δ(t − t′)δij , (5)
and the disorder-dependent equilibrium effective Hamiltonian H:
H =
∫
ddR
{
1
2
[
|∇~ϕ0|
2+r̊|~ϕ0|
2
]
+
v0
4!
|~ϕ0|
4−D(x̂~ϕ0)
2
}
. (6)
In (6), the field ~ϕ0 is an m-component vector, D is proportional to the anisotropy
constant of the spin Hamiltonian (2) with D0; r̊ and v0 are defined by D0 and the
fourth order coupling of the m-vector magnet (see [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 ap-
propriate Lagrangians are studied. For the model equations (4)–(5) the Lagrangian
reads:
L =
∫
ddRdt
∑
i
ϕ̃i,0
[
∂ϕi,0
∂t
+ Γ̊
δH
δϕi,0
− Γ̊ϕ̃i,0
]
(7)
with a new auxiliary response m-component field ~̃ϕ0. Here and below the 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 the quenched disorder, the replica trick
is often used in statics [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 average over the ran-
dom variables x̂ with their distribution (3). Then the Lagrangian for the model reads:
L =
{
∫
ddRdt
∑
i
ϕ̃i,0
[ (
∂
∂t
+Γ̊(r0−∇2)
)
ϕi,0−Γ̊ϕ̃i,0+
Γ̊v0
3!
ϕi,0
∑
j
ϕj,0ϕj,0+
Γ̊y0
3!
ϕ3
i
]
740
Critical slowing down in random anisotropy magnets
+
∫
dt′
∑
i
ϕ̃i,0(t)ϕi,0(t)
[
Γ̊2u0
3!
∑
j
ϕ̃j,0(t
′)ϕj,0(t
′)+
Γ̊2w0
3!
ϕ̃i,0(t
′)ϕi,0(t
′)
]}
. (8)
Here, r0 is proportional to the temperature distance to the mean field critical point
and the bare couplings are u0 > 0, v0 > 0, w0 < 0. Moreover, u0 and w0 are con-
nected to the moments of the distribution (3) in such a way that w0/u0 = −m. The
term with y0 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 {u0, v0, w0, y0} will be denoted below by {u0,i}, i = 1, . . . , 4.
Figure 1. Elements for construction of Feynman diagrams. The response function
and the correlation function read: G(k, ω) = (−iω + Γ̊(r + k2))−1, C(k, ω) =
2Γ̊(| − iω + Γ̊(r + k2)|)−2. A stands for v0/3! · (δi,jδl,m + δi,lδj,m + δi,mδj,l)/3 or
for y0/3! δi,jδj,lδl,m while B is equal to u0/3! δi,jδl,m or to w0/3! δi,jδj,lδl,m .
With the Lagrangian (8) depending on the bare quantities (denoted by the sub-
and 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 function Γ̊i,j
ϕ̃ϕ(r0, {ui,0}, Γ̊, k, ω)
= Γ̊ϕ̃ϕ(r0, {ui,0}, Γ̊, k, ω)δi,j. The calculations are performed using Feynman dia-
grams, elements for them are given in figure 1 whereas the one- and two-loop con-
tributions to Γ̊i,j
ϕ̃ϕ are depicted in figure 2.
We perform renormalization of Γ̊ϕ̃ϕ within the minimal subtraction scheme [3].
In this scheme, in order to define renormalized static (ϕ, r, {ui}) and dynamic (ϕ̃,
Γ) fields and couplings, the renormalization factors Zϕ, Zr, Zui
and Zϕ̃, ZΓ are
introduced by:
ϕ = Z−1/2
ϕ ϕ0, ϕ̃ = Z
−1/2
ϕ̃ ϕ̃0, r = Z−1
r r, ui = µ−εZ−1
ui
Z2
ϕAdu0,i, Γ = ZΓΓ̊. (9)
Here, µ is the external momenta scale, ε = 4 − d, and Ad = Γ(1 − ε/2)Γ(1 +
ε/2)Ωd(2π)−d is a geometrical factor.
741
M.Dudka et al.
k,w
i j
k ’ ’,w k,w
i j
k ’ ’,w
j
k ’ ’,wk,w
i
k,w
i j
k ’ ’,w
j
k ’ ’,wk,w
i
k,w
i j
k ’ ’,w
j
k ’ ’,wk,w
i
k,w
i j
k ’ ’,w j
k ’ ’,wk,w
i
Figure 2. Diagrams of the function Γ̊i,j
ϕ̃ϕ(r0, {ui,0}, Γ̊, k, ω) up to two-loop order.
First two terms represent one-loop contribution, while the rest of the diagrams
are of two-loop order.
The critical behaviour of the system is described by the following RG functions:
βui
({ui}) = µ
∂u
∂µ
∣
∣
∣
∣
0
, ζr({ui}) = −µ
∂ ln Zr
∂µ
∣
∣
∣
∣
0
, ζΓ({ui}) = µ
∂ ln ZΓ
∂µ
∣
∣
∣
∣
0
, (10)
where the symbol
∣
∣
0
means differentiation at fixed bare parameters. The β-functions
determine the RG flow of couplings under renormalization:
`
dui
d`
= βui
({ui}), i = 1, . . . , 4, (11)
and the flow parameter ` is related to the distance from the critical point. Subse-
quently, an information about the critical behaviour of a system can be obtained
from the analysis of the fixed points (FPs) of the flow equations (11). A FP {u∗
i } is
defined as simultaneous zero of all β-functions:
βui
({u∗
i }) = 0, i = 1, . . . , 4. (12)
The stable and accessible FP (from initial conditions) corresponds to the critical
point of the system. A FP is stable if all eigenvalues ωi of the stability matrix
Bi,j = ∂βui
/∂uj 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:
z = 2 + ζΓ(u∗, v∗, w∗, y∗). (13)
While the effective dynamical exponent zeff 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):
zeff = 2 + ζΓ(u(`), v(`), w(`), y(`)) . (14)
We neglect contributions to zeff coming from the amplitude function because they
are considered to be small.
742
Critical slowing down in random anisotropy magnets
3. 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 func-
tion ζΓ 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. In the
present study we use the new two-loop expression for the function ζΓ. The last is
derived from the vertex function Γ̊i,j
ϕ̃ϕ discussed in section 2 and reads:
ζΓ = −
(u + w)
3
+
(6 ln(4/3) − 1)
24
(y2 +
2
3
vy +
m + 2
3
v2)
+
1
36
(
(m + 2)uv + 5u2 + 5w2 + 10uw + 3yw + 3vw + 3uy
)
. (15)
The FP equations (12) solved for the static β-functions at fixed space dimension
d = 3 [37] result in 16 FPs [15,29]. 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)):
– Gaussian FP I: u∗ = v∗ = w∗ = y∗ = 0; z(∀m) = 2;
– pure FP II: v∗ 6= 0, u∗ = w∗ = y∗ = 0; z(m = 2) = 2.053, z(m = 3) = 2.051;
– polymer FP III: u∗ 6= 0, v∗ = w∗ = y∗ = 0; z(∀m) = 1.815;
– Ising FP V: y∗ 6= 0, u∗ = v∗ = w∗ = 0; z(∀m) = 2.052;
– cubic FP VIII: v∗ 6=0, y∗ 6=0, u∗=w∗=0; z(m=2)=2.157, z(m=3)=2.042;
– Ising FP X: y∗ 6= 0, u∗ = −w∗, v∗ = 0; z(∀m) = 2.052;
– random Ising FP XV: w∗ 6= 0, y∗ 6= 0, u∗ = v∗ = 0; z(∀m) = 2.139.
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. However the polymer FP
is not accessible from the physical initial conditions. This leads to the conclusion
[27–30] 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 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 zeff exponent as explained below.
743
M.Dudka et al.
The crossovers between different FPs lead to rich pictures of possible RG flows
[15]. Many flows are effected by the Ising FPs V and X. Introducing into (14) sev-
eral typical RG flows which start from the physical region of initial couplings one
obtains different regimes for approaching the effective dynamical exponent zeff to
asymptotics. The dependences on the flow parameter ` of zeff for easy-plane (m = 2)
and Heisenberg (m = 3) magnets are shown in figures 3 and 4 correspondingly. Flow
3 was chosen to be affected by both Ising FPs V and X. Therefore, both curves 3 of
figures 3 and 4 demonstrate that a large region for zeff might exist with dynamical
exponent values of the pure one-component (Ising) model A. Curves 6 correspond
to the flows which come near the pure FP II and curve 7 in figure 4 corresponds to
the flow which comes near the cubic FP VIII.
-100 -80 -60 -40 -20 0
0,00
0,02
0,04
0,06
0,08
0,10
0,12
0,14
0,16
0,18
0,20
0,22
z
ef
f-2
ln l
3 2
6
1
5
4
XV
V,X
Figure 3. Effective critical exponent zeff as a function of the logarithm of the
flow parameter for order parameter dimension m = 2. Dashed line indicate the
value of z at the FPs V, X. See text for details.
Although the asymptotic exponents of the random anisotropy magnets consi-
dered 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 the smallest static stability exponent ω = −0.0036 [15] which is equal
in absolute value to the ratio of heat capacity critical exponent αr and correlati-
on length critical exponent νr of random-site Ising model [30]. As a consequence
the Wegner correction to scaling is ∆ = ωνr = −αr. 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 crit-
ical behaviour governed by effective critical exponents will be observed experimen-
tally or in the numerical simulations.
As it is seen from figures 3, 4, another particular feature of zeff seems to be that
it always reaches its asymptotic value z from the region zeff > z. Therefore, in an
experimental situation, a decrease of zeff may serve as an evidence of approaching 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
744
Critical slowing down in random anisotropy magnets
-100 -80 -60 -40 -20 0
0,00
0,02
0,04
0,06
0,08
0,10
0,12
0,14
0,16
0,18
0,20
0,22
0,24
4
7
6
3 2
1
5
z
ef
f-2
ln l
XV
V,X
Figure 4. Effective critical exponent zeff as a function of the logarithm of the
flow parameter for the order parameter dimension m = 3. Dashed line indicates
the value of z at the FPs V, X. See the text for details.
of the non-magnetic component, the zeff approaching its asymptotic value is not
necessarily only from the above [38].
4. Conclusions
In this paper, we have analyzed the critical slowing down in magnets effected
by random anisotropy. These magnets have a second order phase transition to the
ferromagnetic 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
ζΓ given in equation (15). Combining this result with the former data for the static
critical behaviour we obtained numerical values for the effective critical exponent
zeff which governs the critical slowing down of the relaxational time when Tc is
approached. In figures 3,4 we give the 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 sys-
tems with cubic distribution is the same as for the random-site Ising systems, the
crossover between different fixed points considerably effects the non-asymptotic criti-
cal properties. Different scenarios of dynamical critical behaviour are observed. Since
asymptotics is approached very slowly, it might be observed in real and numerical
experiments. The effective exponents measured may take values essentially differing
from the asymptotic one (in our calculation z = 2.139). For example in a large re-
gion zeff can be equal to the exponent of the pure Ising model (z = 2.052). Another
particular feature of critical slowing down in random anisotropy magnets which is
predicted by our analysis is that, contrary to the diluted magnets, zeff seems to
always reach its asymptotic value z from the region zeff > z.
745
M.Dudka et al.
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 of 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’s 80th birthday. His early work on the phase
transitions theory and on the non-asymptotic criticality [5] preceded in many re-
spects many later contributions to this field. One of us (Yu.H.) is deeply indebted to
Prof. I.R.Yukhnovskii for introducing him into the fascinating field of phase transi-
tions 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 wissenschaftli-
chen Forschung under Project No. P16574.
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747
M.Dudka et al.
Критичне сповільнення в магнетиках з випадковою
анізотропією
М.Дудка 1,2 , Р.Фольк 2 , Ю.Головач 1,2,3 , Г.Мозер 4
1 Інститут фізики конденсованих систем НАН України,
79011 Львів, Україна
2 Інститут теоретичної фізики, Университет Йогана Кеплера міста
Лінц, A–4040 Лінц, Австрія
3 Львівський національний університет ім. І.Франка, 79005 Львів,
Україна
4 Інститут фізики і біофізики, Університет міста Зальцбург, A–5020
Зальцбург, Австрія
Отримано 18 липня 2005 р.
Ми вивчаємо релаксаційну динаміку з незбережним параметром
порядку (критична динаміка моделі А) для тривимірного магнетика
з безладом у формі випадкової осі анізотропії. Для анізотропного
розподілу випадкових осей асимптотична критична поведінка спів-
падає з поведінкою ізингівських систем з випадковими вузлами.
Таким чином асимптотична критична динаміка керується динаміч-
ним показником випадкової моделі Ізинга. Однак безлад значно
впливає на динамічну поведінку в неасимптотичному режимі. Ми
проводимо теоретико-польовий ренормалізаційно-груповий аналіз
в рамках схеми мінімального віднімання в двопетлевому наближенні
щоб дослідити асимптотичну і ефективну критичну динаміку систем
з випадковою анізотропією. Результати демонструють немонотонну
поведінку динамічного ефективного критичного показника zeff .
Ключові слова: критична динаміка, невпорядковані системи,
випадкова анізотропія, ренормалізаційна група
PACS: 05.50.+q, 05.70.Jk, 61.43.-j, 64.60.Ak, 64.60.Ht
748
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