On the critical behaviour of random anisotropy magnets: cubic anisotropy
The critical behaviour of an m -vector model with a local anisotropy axis of random orientation is studied within the field-theoretical renormalization group approach for cubic distribution of anisotropy axis. Expressions for the renormalization group functions are calculated up to the two-loop o...
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| Опубліковано в: : | Condensed Matter Physics |
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| Дата: | 2001 |
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Інститут фізики конденсованих систем НАН України
2001
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the critical behaviour of random anisotropy magnets: cubic anisotropy / M. Dudka, R. Folk, Yu. Holovatch // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 459-472. — Бібліогр.: 26 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859471204397613056 |
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| author | Dudka, M. Folk, R. Holovatch, Yu. |
| author_facet | Dudka, M. Folk, R. Holovatch, Yu. |
| citation_txt | On the critical behaviour of random anisotropy magnets: cubic anisotropy / M. Dudka, R. Folk, Yu. Holovatch // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 459-472. — Бібліогр.: 26 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | The critical behaviour of an m -vector model with a local anisotropy axis
of random orientation is studied within the field-theoretical renormalization
group approach for cubic distribution of anisotropy axis. Expressions for
the renormalization group functions are calculated up to the two-loop order
and investigated both by an ε = 4 − d expansion and directly at space dimension
d = 3 by means of the Pade-Borel resummation. One accessible ´
stable fixed point indicating a 2nd order ferromagnetic phase transition with
dilute Ising-like critical exponents is obtained.
Критична поведінка m -векторної моделі з локальними осями анізотропії випадкової орієнтації досліджується для кубічного розподілу
осей анізотропії за допомогою методу теоретико-польової ренормалізаційної групи. Вирази для ренормгрупових функцій обчислюються
у двопетлевому наближенні і досліджуються як ε = 4 − d розкладом,
так і безпосередньо при вимірності простору d = 3 пересумовуванням Паде-Бореля. Отримується одна досяжна стійка фіксована точка, яка вказує на фазовий перехід другого роду з критичними показниками розведеної моделі Ізинґа.
|
| first_indexed | 2025-11-24T09:09:10Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 3(27), pp. 459–472
On the critical behaviour of random
anisotropy magnets: cubic anisotropy
M.Dudka 1 , R.Folk 2 , Yu.Holovatch 1,3
1 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2 Institut für Theoretische Physik, Johannes Kepler Universität Linz,
A-4040 Linz, Austria
3 Ivan Franko National University of Lviv,
12 Drahomanov Str., 79005 Lviv, Ukraine
Received April 2, 2001, in final form July 13, 2001
The critical behaviour of an m -vector model with a local anisotropy axis
of random orientation is studied within the field-theoretical renormalization
group approach for cubic distribution of anisotropy axis. Expressions for
the renormalization group functions are calculated up to the two-loop order
and investigated both by an ε = 4 − d expansion and directly at space di-
mension d = 3 by means of the Padé-Borel resummation. One accessible
stable fixed point indicating a 2nd order ferromagnetic phase transition with
dilute Ising-like critical exponents is obtained.
Key words: random anisotropy, renormalization group, critical exponents
PACS: 61.43.-j, 64.60.Ak
1. Introduction
In statistical physics, low-temperature phases of many-particle spin models may
possess qualitatively different features depending on the fact whether the corre-
sponding spin Hamiltonian is of discrete or of continuous symmetry. An m-vector
model described by the Hamiltonian [1] may serve as a textbook example:
H = −
∑
R,R′
JR,R′
~SR
~SR′, (1)
where vectors R,R′ span sites of the d-dimensional (hyper)cubic lattice, JR,R′ > 0
is a short-range ferromagnetic interaction and ~SR
~SR′ is a scalar product of classical
m-component “spins” ~SR = (S1
R′, . . . , Sm
R′). The Hamiltonian (1) possesses a global
O(m) symmetry: it remains invariant under rotations in the space of vectors { ~S}.
c© M.Dudka, R.Folk, Yu.Holovatch 459
M.Dudka, R.Folk, Yu.Holovatch
However, continuous O(m) symmetry turns to the discrete one for m = 1: this
corresponds to the invariance of the Ising model Hamiltonian under discrete turn
of the Ising spins to opposite direction. The consequences are well known: whereas
in the Ising model a ferromagnetic phase exists for lattice dimensions greater than
1 [1] (i.e. the lower critical dimension dL = 1), dL = 2 for m > 2 [2]. For d > 2 a
continuous symmetry can be spontaneously broken for any m, while dU = 4 is the
upper critical dimension of the m-vector model: starting from d = 4 the magnetic
phase transition is governed by the mean-field critical exponents.
Implementation of a weak structural (lattice) disorder into model (1) has crucial
consequences for the existence of the ordered phase. For a particular reason of the
present study of main interest to us will be the case when distribution of disorder
can be characterized by a certain symmetry. The m-vector model with random
anisotropy (a random anisotropy model, RAM) [3] may serve as an example:
H = −
∑
R,R′
JR,R′
~SR
~SR′ −D0
∑
R
(x̂R
~SR)
2. (2)
Here, the notations are as in (1), D0 > 0 is an anisotropy strength, and x̂R is an unit
vector pointing in the local (quenched) random direction of an uniaxial anisotropy.
The model Hamiltonian (2) possesses randomness only for m > 1: at m = 1
the second term is a constant and leads to a shift in a free energy of the resulting
regular (Ising) model. It means that the randomness-induced behaviour in RAMmay
be observed only for spins of continuous symmetry. The low-temperature ordering
in RAM is also influenced in addition to the global variables of a regular magnet
(i.e. lattice dimension, type of interaction and spin symmetry) also by distribution
of the random variables x̂ ≡ x̂R in (2). For non-correlated x̂R the low-temperature
ordering depends on the probability distribution p(x̂) of the direction of anisotropy
on a single site. In particular, for the isotropic distribution dL = 4: a ferromagnetic
order is absent for the lattice dimension less than 4. The absence of ferromagnetic
ordering in isotropic RAM was first observed in the renormalization group study
of [4] where no accessible fixed points of the renormalization group transformation
were obtained for the model within ε = 4 − d expansion. Recently, this result was
corroborated by higher-order calculations refined by a resummation technique [5].
The proof of [6,7] used arguments similar to those applied by Imry and Ma [8] for
a random-field Ising model and showed that the susceptibility of the ordered state
diverges for d < 4. Explicit calculations for m → ∞ were offered in [6]. Although the
latter appeared to be erroneous [9], the value dL = 4 was further supported by an
attempt of a Mermin-Wagner proof of the absence of ferromagnetism in RAM with
the isotropic distribution of an anisotropy axis for d < 4 [10]. The proof of the work
[10] uses the replica trick [11] and cannot be considered rigorous. However, the same
paper studies RAM at low temperatures and thus the small anisotropy that avoids
the application of replicas by means of the Migdal-Polyakov renormalization group
technique and the fixed point structure obtained there in d− 4 dimensions confirms
the absence of ferromagnetism below d = 4. The upper critical dimension for RAM
with the isotropic distribution of an anisotropy axis was shown to be dU = 6 [12].
460
Critical behaviour of random anisotropy magnets
However, the above arguments do not concern anisotropic distributions p(x̂).
Here, the possibility of ferromagnetic ordering is to be studied for every particular
case. We address the question of the existence of a ferromagnetic second order phase
transition and its universal properties for a d = 3 RAM with an anisotropic distri-
bution of a random axis, when the vector x̂R (1) points only along one of the 2m
directions of axes k̂i of a cubic lattice (the so-called cubic anisotropy):
p(x̂) =
1
2m
m
∑
i=1
[δ(m)(x̂− k̂i) + δ(m)(x̂+ k̂i)], (3)
where δ(ŷ) are Kronecker’s deltas. Besides a pure academic interest, such a choice
has practical applications: typical examples of random-anisotropy magnets are amor-
phous rare-earth – transition metal alloys [13] and the cubic distribution (3) of a
random axis mimics the situation when an amorphous magnet still “remembers”
initial (cubic) lattice structure.
In the present paper, we study the RAM with the cubic distribution of a random-
anisotropy axis (3) by means of a field theoretical renormalization group (RG) tech-
nique [14] and analyze the two-loop RG functions both by an ε = 4 − d expansion
and directly at space dimension d = 3. We show the existence of a second order
phase transition and make our conclusions about its numerical characteristics based
on the resummation technique applied to the resulting perturbation theory series.
The paper is a direct continuation of our preceding work [5], where we applied sim-
ilar tools to study RAM with an isotropic distribution p(x̂) and we refer the reader
there for a more extended review of the RAM general features.
The set-up of the paper is the following: in the next section 2 we describe the
model and obtain the RG functions within the massive field theory scheme. In the
RG analysis, the presence of a second order phase transition corresponds to the
presence of a reachable stable fixed point of the RG transformation. The fixed points
and their stability are analyzed in section 3 by means of an ε-expansion to order ε2
and by resummation of a d = 3 series. We estimate the critical exponents values and
display them in section 3 as well. Section 4 concludes our study and summarizes the
results obtained.
2. The renormalization group functions
In order to apply the field theoretical RG approach to study the critical behaviour
of the RAM (2) with quenched local anisotropy axis distributed according to (3) one
should get an effective Hamiltonian of the model. Following the scheme of [4] for a
given configuration of quenched random variables x̂R in (2) the partition function
of RAM is written in the form of a functional integral of a Gibbs distribution with
the effective Hamiltonian:
H(x̂R, ~φ) = −
∫
ddR
{
1
2
[
r0|~φ|2+|~∇~φ|2
]
−D1(x̂R
~φ)2+v0|~φ|4+ . . .
}
, (4)
461
M.Dudka, R.Folk, Yu.Holovatch
where D1 is proportional to D0, r0 and v0 are defined by D0 and the familiar bare
couplings of an m-vector model, and ~φ ≡ ~φR is a m-dimensional vector. Implying
quenched disorder and using therefore the replica trick [11] one arrives at the n-
replicated configuration-dependent partition function. Performing then the average
over random variables for the case of a cubic distribution (3) one ends up with the
effective Hamiltonian [4]:
Heff = −
∫
ddR
{
1
2
[
µ0
2|~ϕ|2+|~∇~ϕ|2
]
+u0|~ϕ|4+v0
n
∑
α=1
|~φα|4
+ w0
m
∑
i=1
n
∑
α,β=1
(φα
i )
2(φβ
i )
2+y0
m
∑
i=1
n
∑
α=1
(φα
i )
4
}
, (5)
which in the replica n → 0 limit describes critical behaviour of the model (2) with
cubic anisotropy distribution (3). Here, µ0 is bare mass, bare couplings u0 > 0,
v0 > 0, w0 < 0 are defined by D0 and familiar bare couplings of an m-vector model.
The last term in (5) combines the symmetries of terms with coefficients v 0 and w0. It
does not result from the functional representation of the free energy but is generated
by further application of the RG transformation. Therefore y0 can be of either sign.
φα
i are the components of anmn-dimensional order parameter field, |ϕi|2 =
∑
α |φα
i |2.
Values w0 and u0 are related to appropriate cumulants of the distribution function
(3) in such a way that their ratio w0/u0 = −m and thus determines a region of
typical initial values in the (u− v − w − y)-space of couplings.
To get a qualitative picture of a critical behaviour it is standard now to rely on
the field-theoretical RG approach [14]. In this approach, finiteness of the (renormal-
ized) vertex functions Γ
(n)
R is ensured by imposing certain normalizing conditions. In
turn, this leads to different renormalization schemes. Here, we will make use of the
renormalization at fixed mass and zero external momenta {k} [15]. Normalization
conditions are written then for a fixed space dimension d and read:
Γ
(2)
R (0;µ2, u, v, w, y) = µ2,
∂
∂k2
Γ
(2)
R (k;µ2, u, v, w, y)
∣
∣
∣
k2=0
= 1,
Γ(4)
u R({0};µ2, u, v, w, y) = µ4−du,
Γ(4)
v R({0};µ2, u, v, w, y) = µ4−dv,
Γ(4)
w R({0};µ2, u, v, w, y) = µ4−dw,
Γ(4)
y R
({0};µ2, u, v, w, y) = µ4−dy;
Γ
(2,1)
R (p; k;µ2, u, v, w, y)
∣
∣
∣
p2=k2=0
= 1. (6)
Here, µ, u, v, w, y are renormalized mass and dimensionless couplings, Γ
(2,1)
R is renor-
malized vertex function with φ2 insertion, and the vertices Γ
(4)
u , Γ
(4)
v ,Γ
(4)
w , Γ
(4)
y are
parts of a full vertex function
Γ(4)i1i2i3i4
α1α2α3α4
= Γ(4)
u Si1i2i3i4
α1α2α3α4
+ Γ(4)
v Si1i2i3i4Fα1α2α3α4
+ Γ(4)
w Fi1i2i3i4Sα1α2α3α4
462
Critical behaviour of random anisotropy magnets
+ Γ(4)
y Fi1i2i3i4Fα1α2α3α4
, (7)
where
Fijkl = δijδikδil,
Sijkl =
1
3
(δijδkl + δikδjl + δilδjk) ,
Sαβγτ
ijkl =
1
3
(δijδklδαβδγτ+ δikδjlδαγδβτ+ δilδjkδατδβγ) , (8)
δab is Kronecker’s delta. Tensors in (8) correspond to terms of different symmetry in
the effective Hamiltonian (5), the Latin symbols are the spin indices and the Greek
symbols are the replica indices.
The mass is renormalized by: µ = ZφΓ
(2)(0;µ0; {ui,0}), with ui = u, v, w, y and
Zφ being the field renormalizing factor. Finiteness of Γ(2,1) is secured by the factor
Z̄φ2. The renormalizing factors of couplings Zui
relate the bare couplings to the
renormalized ones
ui,0 = µ4−dZui
Z2
φ
ui. (9)
All renormalizing factors are defined by conditions (6). A change of the couplings u i
and Z-factors under the RG transformation is described by the β- and γ-functions:
βui
=
∂ui
∂ lnµ
, γφ =
∂ lnZφ
∂ lnµ
, γ̄φ2 = −∂ ln Z̄φ2
∂ lnµ
, (10)
determining the approach of the system to criticality. Namely, the fixed point (FP)
{u∗
i } of the RG transformation defined as a solution of equations
βui
({u∗
j}) = 0 (11)
if stable, may correspond to the critical point. The condition of the FP stability
reads:
∣
∣
∣
∣
∂βui
∂uj
({u∗
j})− ωiδij
∣
∣
∣
∣
= 0, Re ωi > 0. (12)
However, the correspondence of a stable FP to the critical point of a system implies
that this FP is reachable from the initial conditions (initial values of the couplings).
In the stable FP, the correlation length critical exponent and the pair correlation
function critical exponent are defined by:
ν−1 = 2− γ̄φ2({u∗
i })− γφ({u∗
i }), (13)
η = γφ({u∗
i }). (14)
The rest of critical exponents may be derived from the familiar scaling relations.
Applying the renormalization scheme (6) we get the RG functions of the model
(5) in a two-loop approximation. In the replica limit n = 0 they read:
βu = −ε
{
u− 1
6
[
8u2+2 (m+2)uv+2vw+4uw+6uy
]
+
1
9
[
44u3+48u2w
463
M.Dudka, R.Folk, Yu.Holovatch
+12w2u+24 (m+ 2) vu2+2 (3m+6)uv2+4w2v+4v2w+72u2y+18y2u
+60uvw+36uvy+36uwy
]
i1 +
2
9
[
2u3+2 (m+2) vu2+(m+2)uv2
+2w2u+6uvw+4u2w+6u2y+6uvy+6uwy+3y2u
]
i2
}
, (15)
βv = −ε v
{
1− 1
6
[(m+8) v+12u+4w+6y] +
1
9
[
2 (5m+22) v2+8 (3m+15) vu
+84u2+12w2+68vw+72vy+18y2+72uw+108uy+36wy
]
i1+
2
9
[
(m+2)v2
+2 (m+ 2)uv+2u2+2w2+6vw+4uw+6uy+6vy+6wy+3y2
]
i2
}
, (16)
βw = −ε w
{
1− 1
6
[8w+12u+4v+6y] +
1
9
[
44w2+84u2+120wu+2 (m+6) v2
+68vw+72wy+18y2+2 (6m+ 36)uv+108uy+36vy
]
i1+
2
9
[
(m+2)v2
+2 (m+ 2)uv+2u2+2w2+6vw+4uw+6uy+6vy+6wy+3y2
]
i2
}
, (17)
βy = −ε
{
y − 1
6
[
9y2+8vw+12uy+12vy+12wy
]
+
1
9
[
(4m+ 72) v2w+72w2v
+54y3+84u2y+(6m+ 84) v2y+84w2y+144y2u+144y2v+144y2w
+96uvw+2 (6m+ 84)uvy+252vwy+168uwy
]
i1+
2
9
[
2u2y+2 (m+ 2) uvy
+ (m+2) v2y+2w2y+6vwy+4uwy+6y2u+6y2v+6y2w+3y3
]
i2
}
, (18)
γφ = −ε
9
(
2u2 + 2 (m+ 2)uv + (m+ 2)v2 + 6vw + 2w2 + 4uw + 6uy
+6vy + 6wy + 3y2
)
i2, (19)
γ̄φ2 =
ε
3
{
1
2
(
2u+ (m+ 2) v + 2w + y
)
−
(
2u2 + 2 (m+ 2) uv + (m+ 2) v2
+6vw + 2w2 + 4uw + 6uy + 6vy + 6wy + 3y2
)
i1
}
, (20)
where ε = 4 − d and i1, i2 are loop integrals [16] of the diagrams
r ✍✌
✎☞
✟
✟✟
❍
❍❍
r
r❍❍
✟✟
and
∂
∂k2
✍✌
✎☞rr
|k2=0 correspondingly. Analysing the RG functions (15)–(20) in the fixed
d = 3 scheme [15] one substitutes the loop integrals by their numerical values i1 (d =
3) = 1/6 , i2(d = 3) = −2/27 [17] and then deals with the expansions (15)–(20) in
renormalized couplings. However, the ε-expansion technique [18] is also applicable
464
Critical behaviour of random anisotropy magnets
to the massive scheme. To this end, the loop integrals are to be substituted by their
ε-expansion: i1 ≃ 1/2 + ε/4 + . . ., i2 ≃ −ε/8 + . . . [19] and the perturbation theory
is constructed both in ε and in renormalized couplings. Both schemes will be used
in our analysis of the expressions (15)–(20) in the next section.
To conclude this section, let us note that the one-loop parts of the RG functions
(15)–(20) reproduce ε-expansion results of works [4,20]. Moreover, the RAM with
the cubic distribution of random axis may be regained from the more general model
of the article [21] describing phase transition in crystals with low-symmetry point
defects. There, the corresponding RG functions were written to order ε2. The ε–
expansion of RG functions (15)–(20) does not coincide with the appropriate functions
of the work [21] as far as the renormalization schemes differ. However, as it will be
shown in the next section, the observables obtained in ε-expansion on their basis do
coincide, as expected.
3. The fixed points and the critical exponents
3.1. An ε -expansion
The first RG study of the RAM [4] reported an evidence of 14 FPs for the cubic
distribution of an anisotropy axis. This result was obtained in the first order in ε
and may be easily reproduced based on the functions (15)–(18) putting two-loop
contributions equal to zero and applying the ε-expansion scheme as described at the
end of section 2. We list coordinates of the FPs I–XIV in the table 1 (in order to
recover the results of [4] we extract the value of the one-loop integral ∼ 1/ε from
conventionally normalized couplings: see note [16]).
The results of linear in ε analysis [4] state that among the FPs with u > 0, v >
0, w < 0 only a “polymer” O(n = 0) FP III is stable at all m for ε > 0, but it
is not reachable from the initial values of couplings (see figure 1). The reason is
a separatrix joining the unstable FPs I and VII and separating initial values of
couplings (shown by a cross in figure 1) and FP III. Possible runaway behaviour of
RG flows lead Aharony [4] to the conclusion about smearing of the phase transition
as Tc approaches.
However, the subsequent study of Mukamel and Grinstein [20] brought about a
possibility of a second order phase transition with the scenario of a weakly diluted
quenched Ising model [22]. Indeed, performing perturbation theory expansion to the
order ε2 we get not only the corrections to the coordinates of the FPs I–XIV (listed
in the tables 1, 2) but the new FPs XV, XVI, XVII (see the bottom of the table 1).
The appearance of the pairs of the FPs XV and XVI is caused by the well known
fact that the β-functions βw, βy at u = v = 0 (βu, βv at w = y = 0, correspondingly)
are degenerated at the one loop level. Expressions of FPs coordinates XV, XVI in
the table 1 are a familiar
√
ε expansion of the FP of weakly diluted quenched Ising
model [22]. The
√
ε expansion of the FP XVII holds for m = 2 and is caused by the
one-loop degeneracy of the βu, βv, βy functions for w = 0 (c.f. singularity at m = 2
in the ε-expansion of the FP IX).
465
M.Dudka, R.Folk, Yu.Holovatch
Table 1. FPs of the RAM with the random cubic anisotropy distribution (ε-
expansion). Here, α± = (m− 4±
√
m2 + 48)/8, β± = −(m+ 12±
√
m2 + 48)/6,
A±± = 6α± + 3β± + m + 6. Note, that the fixed points XV–XVII appear only
in the two-loop approximation due to the degeneracy of the corresponding one-
loop functions. Expressions for some two-loop contributions (indexed by Roman
numbers) are too cumbersome, their numerical values are listed in table 2 for
some m.
u∗ v∗ w∗ y∗
I. 0 0 0 0
II. 0 6
m+8
ε+18 (3m+14)
(m+8)3
ε2 0 0
III. 6
8
ε+ 63
128
ε2 0 0 0
IV. 0 0 6
8
ε+ 63
128
ε2 0
V. 0 0 0 6
9
ε+ 34
81
ε2
VI. 6(m−4)
16(m−1)
ε+uVIε
2 6
4(m−1)
ε+ vVIε
2 0 0
VII. 3
2
ε+ 3
4
ε2 0 −3
2
ε− 3
4
ε2 0
VIII. 0 2
m
ε+ vVIIIε
2 0 2(m−4)
3m
ε+yVIIIε
2
IX. m−4
4(m−2)
ε+uIXε
2 1
m−2
ε+ vIXε
2 0 m−4
3(m−2)
ε+ yIXε
2
X. 1
2
ε+ 25
108
ε2 0 −1
2
ε− 25
108
ε2 2
3
ε+ 34
81
ε2
XI.α+β+
3α+
A++
ε+ uXIε
2 3
A++
ε+ vXIε
2 3(m+4)
4A++
ε+ wXIε
2 3β+
A++
ε+ yXIε
2
XII.α+β−
3α+
A+−
ε+ uXIIε
2 3
A+−
ε+ vXIIε
2 3(m+4)
4A+−
ε+ wXIIε
2 3β−
A+−
ε+ yXIIε
2
XIII.α−β+
3α−
A−+
ε+ uXIIIε
2 3
A−+
ε+ vXIIIε
2 3(m+4)
4A−+
ε+wXIIIε
2 3β+
A−+
ε+ yXIIIε
2
XIV.α−β−
3α−
A−−
ε+ uIVε
2 3
A−−
ε+ vIVε
2 3(m+4)
4A−−
ε+ wIVε
2 3β−
A−−
ε+ yIVε
2
XV. 0 0 ∓
√
54
53
ε ±4
3
√
54
53
ε
XVI. ∓
√
54
53
ε 0 0 ±4
3
√
54
53
ε
XVII.m = 2 ±
√
54
53
ε ∓2
√
54
53
ε 0 ±4
3
√
54
53
ε
466
Critical behaviour of random anisotropy magnets
Checking the stability of new FPs XV–XVIII we find that all of them are unstable
except for the FP with w < 0, y > 0 from the pair XV. Moreover, this point is
reachable from the initial values of the couplings. As far as it is the FP of the diluted
Ising model one concludes, that in the critical region, RAM with cubic distribution
of random anisotropy axis (3) decouples into m independent dilute Ising models
and the phase transition is governed by the familiar random Ising model critical
exponents [23].
However, let us keep in mind that the above picture is obtained in the frames of
the “naive” analysis of ε (and
√
ε) expansion and it is highly desirable to confirm it
by a more reliable analysis of FPs and their stability. This will be done below.
3.2. A d = 3 series
Table 2. Numerical values of some contributions to the fixed points coordinates
of the table 1 for m = 2, 3, 4.
m uIV uIX uXI uXII uXIII uXIV
2± –3.8906 ±∞ –29.0018 0.2158 0.0598 0.3838
3 –0.6665 1.2133 –5.3683 0.2319 –0.0161 –9.5594
4± –0.2292 0.5 –2.1797 0.2433 –0.0120 ±∞
vIV vVIII vIX vXI vXII vXIII
2± 3.2578 0.6296 ∓∞ –29.4605 0.1785 0.0372
3 0.8346 0.2689 –1.6030 –2.9857 0.1277 0.1395
4 0.5 0.1042 –0.3958 –0.7266 0.0923 0.2959
vXIV wXI wXII wXIII wXIV yVIII
2 –0.0627 –54.2734 0.2556 –0.1332 –0.2614 –0.4198
3 10.3646 –8.8086 0.2288 –0.0863 18.5654 0.1079
4± ∓∞ –3.5156 0.2054 0.0762 ∓∞ 0.3333
yIX yXI yXII yXIII yXIV
2± ±∞ 135.4989 –0.1911 0.4562 0.2200
3 0.9788 20.9753 –0.1543 0.3749 –12.5612
4± 0.3333 8.1563 0.1231 0.1289 ±∞
The next step in our analysis will be to consider the series (15)–(20) for the RG
functions directly at fixed space dimension d = 3. In the field theory, expansions in
renormalized couplings are known to be asymptotic at best and certain resummation
procedure is needed in order to obtain reliable data on their basis. Here, we will make
use of the Padé-Borel resummation techniques [24]. It consists in the following steps.
For the given initial polynomial in several (in our case in four) variables for any series
of β = βui
from the expressions (15)–(18)
β(u, v, w, y) =
∑
16i+j+k+l63
ai,j,k,l u
ivjwkyl (21)
467
M.Dudka, R.Folk, Yu.Holovatch
one introduces a “resolvent” polynomial [25] in one auxiliary variable λ by:
F (u, v, w, y;λ) =
∑
16i+j+k+l63
ai,j,k,l u
ivjwkylλi+j+k+l−1. (22)
with obvious relation F (u, v, w, y;λ = 1) = β(u, v, w, y). Then, the Borel image of
(22) is defined as:
FB(u, v, w, y;λ) =
∑
16i+j+k+l63
ai,j,k,lu
ivjwkylλi+j+k+l−1
(i+ j + k + l − 1)!
. (23)
Truncated series (23) is approximated by Padé-approximant [1/1](λ). Then the re-
summed β-function is obtained from the formula:
βres(u, v, w, y) =
∫
∞
0
dt exp(−t)[1/1](t). (24)
Similar technique is used for resummation of the expression ν−1 = 2 − γφ({u∗
i}) −
γ̄φ2({u∗
i }). The pair correlation function critical exponent η is obtained by a direct
substitution of FPs values into (14).
✇ ✇
✇
w
u
❵✇
y
✇
✇
I
V
XV X
III
VII
Figure 1. Fixed points of the RAM with cubic distribution of a local anisotropy
axis for v = 0. The only FPs located in the region u > 0, w < 0 are shown. Filled
boxes show the stable FPs, a cross denotes the region of the typical initial values
of couplings.
Applying the resummation procedure (22)–(24) to the β-functions (15)–(18) we
get 16 FPs. In table 3 we present numerical values of FPs coordinates with u∗ > 0,
468
Critical behaviour of random anisotropy magnets
v∗ > 0, w∗ < 0. We visualize the FP picture in figure 1 for v = 0. The last FP XV in
table 3 corresponds to the stable FP of
√
ε-expansion of pair XV in table 1. It has
coordinates with u∗ = v∗ = 0, w∗ < 0 and y∗ > 0 and is accessible from the typical
initial values of couplings (marked by a cross in figure 1).
Table 3. Resummed values of the FPs and critical exponents for the cubic distri-
bution in the two-loop approximation at d = 3. Only FPs with u∗ > 0, v∗ > 0,
w∗ < 0 are shown. The only FPs III and XV are stable.
FP m u∗ v∗ w∗ y∗ ν η
I ∀m 0 0 0 0 1/2 0
2 0 0.9107 0 0 0.663 0.027
II 3 0 0.8102 0 0 0.693 0.027
4 0 0.7275 0 0 0.720 0.026
III ∀m 1.1857 0 0 0 0.590 0.023
V ∀m 0 0 0 1.0339 0.628 0.026
VI 3 0.1733 0.6460 0 0 0.659 0.027
4 0.2867 0.4851 0 0 0.653 0.027
VII ∀m 2.1112 0 –2.1112 0 1/2 0
2 0 1.5508 0 –1.0339 0.628 0.026
VIII 3 0 0.8393 0 –0.0485 0.693 0.027
4 0 0.5259 0 0.3624 0.709 0.026
IX 3 0.1695 0.7096 0 –0.1022 0.659 0.027
4 0.2751 0.4190 0 0.1432 0.653 0.027
X ∀m 0.6678 0 –0.6678 1.0339 0.628 0.026
XV ∀m 0 0 –0.4401 1.5933 0.676 0.031
Applying the resummation procedure (22)–(24) we have not found any other
stable FPs in the region of interest.Thus we are drawn to the conclusion that the
effective Hamiltonian (5) in critical regime reduces to a product ofm effective Hamil-
tonians of a weakly diluted quenched random site Ising model. This means that for
any value of m the system is characterized by the same set of critical exponents
which are those of a weakly diluted random site quenched Ising model.
In the other FPs, we recover the familiar two-loop numerical results for the
Gaussian (FPs I, VII), m-vector (FP II), polymer O(n = 0) (FP III), Ising (FPs
V, X), diluted m-vector (FP VI), and cubic (FP VIII) models. FP IX belongs to
the new universality class. In table 3 we give the numerical values of the critical
exponents in these FPs as well: if the flow from the initial values of couplings pass
near these FPs, one may observe an effective critical behaviour governed by these
critical exponents [23].
469
M.Dudka, R.Folk, Yu.Holovatch
4. Conclusions
In this paper, we presented an analysis of an m-vector model with quenched dis-
order of a random anisotropy type as described by the Hamiltonian (2). It possesses
randomness only for m > 1 and the randomness-induced behaviour in RAM may be
observed only for spins of continuous symmetry. We were interested in a possibility
of a ferromagnetic ordering of RAM for certain anisotropic distribution of a random
anisotropy axis. In particular, we studied the case when the local anisotropy axis
points along the edges of an m-dimensional hypercube.
We used the field theoretical RG approach, obtaining RG functions in the two-
loop approximation and analysing them both by an ε-expansion as well as by re-
summation of the expansion for fixed space dimension d = 3. In the RG language,
the critical point of a system corresponds to the accessible stable FP of the RG
transformation. In our analysis, we get two stable FPs. One of them (FP III in
figure 1) is not accessible for the flows from the region of initial values of couplings,
but the other one FP XV may be reached from these values. Taken that the FP
XV is of the random site Ising type, we conclude that RAM with cubic distribution
of random anisotropy axis is governed by a set of critical exponents of a weakly
diluted quenched Ising model [23]. There is a simple physical interpretation of the
phenomena observed: since the m easy axes of RAM with cubic distribution are
mutually orthogonal, a spin oriented along a given axis feels only the presence of
near-neighbour spins constrained to lie upon the same axis. The system, therefore,
decomposes into m independent diluted Ising models [20,21,26]. Note once more,
that this behaviour is characteristic only of RAM with cubic distribution of ran-
dom anisotropy axis, described by the effective Hamiltonian (5). A distribution of
random anisotropy axis is relevant, i.e., for isotropic distribution, all investigations
bring about an absence of a second order phase transition for d 6 4 [4–10,12].
To conclude we want to attract attention to a certain similarity in the critical
behaviour of both random-site [22] and random-anisotropy [3] quenched magnets: if
at all there appears any new critical behaviour it always is governed by critical expo-
nents of site-diluted Ising type. Thus, in a random-anisotropy system the situation
may occur that the critical behaviour of a system of spins of continuous symmetry
is the same as that of a random-site system with discrete (Ising) spins. The above
calculations of a critical behaviour of RAM were based on a two-loop expansion
improved by a resummation technique. Once the qualitative picture became clear
there is no need to go into higher orders of a perturbation theory as far as the critical
exponents of the site-diluted Ising model are known by now with high accuracy [23].
As a possible generalization of the RAM one may consider a case when quenched
randomness is present in both random-site and random-anisotropy forms. Then, one
arrives [4] to the effective Hamiltonian (5) where the coupling u0 may be of either
sign. We have checked the region u < 0 for the presence of new FPs and verified
that they are absent. Therefore, again FP XV is the only one reachable stable FP
and the observed critical behaviour is unique.
This work has been supported in part by “Österreichische Nationalbank Jubi-
470
Critical behaviour of random anisotropy magnets
läumsfonds” (Austria) through the grant No. 7694 and by the MNTC “Ukryttia”
(Ukraine) through project No. 02/2001.
References
1. see e.g. Stanley H. Introduction to Phase Transitions and Critical Phenomena. Oxford,
Claredon Press, 1971.
2. Absence of the phase transition with spontaneous order parameter for the systems with
continuous symmetry and a short range interaction at d = 2 follows from the Mermin-
Wagner theorem: Mermin N.D., Wagner N. // Phys. Rev. Lett., 1966, vol. 17, p. 1133;
exact solutions for 2d classical O(m) model for m = 4 and m = 3 also demonstrate
the absence of magnetic ordering: Polyakov A.M., Wiegman P.B. // Phys. Lett., 1983,
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11. Emery V.J. // Phys. Rev. B, 1975, vol. 11, p. 239.
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14. Amit D.J. Field Theory, the Renormalization Group, and Critical Phenomena. Sin-
gapore, World Scientific, 1984; Zinn-Justin J. Quantum Field Theory and Critical
Phenomena. Oxford, Oxford University Press, 1989; Kleinert H., Schulte-Frohlinde V.
Critical Properties of φ4-Theories. Singapore, World Scientific, 2001.
15. Parisi G. 1973 (unpublished); J. Stat. Phys., 1980, vol. 23, p. 49.
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loop integrals.
17. Nickel B.G., Meiron D.I., Baker G.A. Jr. // Univ. of Guelph Report, 1977 (unpub-
lished).
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19. Brezin E., Le Guillou J.C., Zinn-Justin J. // Phys. Rev. D, 1973, vol. 8, p. 434.
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21. Korzhenevskii A.L., Luzhkov A.A. // Sov. Phys. JETP, 1988, vol. 67, p. 1229.
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23. See e.g. Folk R., Holovatch Yu., Yavors’kii T. // Phys. Rev. B, 2000, vol. 61, p. 15114
for a recent review on random d = 3 Ising model.
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the random Ising model critical exponents in this case.
471
M.Dudka, R.Folk, Yu.Holovatch
Критична поведінка магнетиків із випадковою
анізотропією: кубічна анізотропія
М.Дудка 1 , Р.Фольк 2 , Ю.Головач 1,3
1 Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
2 Інститут теоретичної фізики, Університет Йоганна Кеплера в Лінці,
A-4040 Лінц, Австрія
3 Львівський національний університет ім. І.Фpанка, 79005 Львів
Отримано 2 квітня 2001 р., в остаточному вигляді – 13 липня
2001 р.
Критична поведінка m -векторної моделі з локальними осями анізо-
тропії випадкової орієнтації досліджується для кубічного розподілу
осей анізотропії за допомогою методу теоретико-польової ренорма-
лізаційної групи. Вирази для ренормгрупових функцій обчислюються
у двопетлевому наближенні і досліджуються як ε = 4− d розкладом,
так і безпосередньо при вимірності простору d = 3 пересумовуван-
ням Паде-Бореля. Отримується одна досяжна стійка фіксована точ-
ка, яка вказує на фазовий перехід другого роду з критичними показ-
никами розведеної моделі Ізинґа.
Ключові слова: випадкова анізотропія, ренормалізаційна група,
критичні показники
PACS: 61.43.-j, 64.60.Ak
472
|
| id | nasplib_isofts_kiev_ua-123456789-120456 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-11-24T09:09:10Z |
| publishDate | 2001 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Dudka, M. Folk, R. Holovatch, Yu. 2017-06-12T07:52:42Z 2017-06-12T07:52:42Z 2001 On the critical behaviour of random anisotropy magnets: cubic anisotropy / M. Dudka, R. Folk, Yu. Holovatch // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 459-472. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 61.43.-j, 64.60.Ak DOI:10.5488/CMP.4.3.459 https://nasplib.isofts.kiev.ua/handle/123456789/120456 The critical behaviour of an m -vector model with a local anisotropy axis of random orientation is studied within the field-theoretical renormalization group approach for cubic distribution of anisotropy axis. Expressions for the renormalization group functions are calculated up to the two-loop order and investigated both by an ε = 4 − d expansion and directly at space dimension d = 3 by means of the Pade-Borel resummation. One accessible ´ stable fixed point indicating a 2nd order ferromagnetic phase transition with dilute Ising-like critical exponents is obtained. Критична поведінка m -векторної моделі з локальними осями анізотропії випадкової орієнтації досліджується для кубічного розподілу осей анізотропії за допомогою методу теоретико-польової ренормалізаційної групи. Вирази для ренормгрупових функцій обчислюються у двопетлевому наближенні і досліджуються як ε = 4 − d розкладом, так і безпосередньо при вимірності простору d = 3 пересумовуванням Паде-Бореля. Отримується одна досяжна стійка фіксована точка, яка вказує на фазовий перехід другого роду з критичними показниками розведеної моделі Ізинґа. This work has been supported in part by “Osterreichische Nationalbank Jubiläumsfonds” (Austria) through the grant No. 7694 and by the MNTC “Ukryttia” (Ukraine) through project No. 02/2001. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the critical behaviour of random anisotropy magnets: cubic anisotropy Критична поведінка магнетиків із випадковою анізотропією: кубічна анізотропія Article published earlier |
| spellingShingle | On the critical behaviour of random anisotropy magnets: cubic anisotropy Dudka, M. Folk, R. Holovatch, Yu. |
| title | On the critical behaviour of random anisotropy magnets: cubic anisotropy |
| title_alt | Критична поведінка магнетиків із випадковою анізотропією: кубічна анізотропія |
| title_full | On the critical behaviour of random anisotropy magnets: cubic anisotropy |
| title_fullStr | On the critical behaviour of random anisotropy magnets: cubic anisotropy |
| title_full_unstemmed | On the critical behaviour of random anisotropy magnets: cubic anisotropy |
| title_short | On the critical behaviour of random anisotropy magnets: cubic anisotropy |
| title_sort | on the critical behaviour of random anisotropy magnets: cubic anisotropy |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120456 |
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