Behaviour of the order parameter of the simple magnet in an external field
The effect of a homogeneous external field on the three-dimensional uniaxial magnet behaviour near the critical point is investigated within the framework of the nonperturbative collective variables method using the ρ⁴ model. The research is carried out for the low-temperature region. The anal...
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Інститут фізики конденсованих систем НАН України
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| Cite this: | Behaviour of the order parameter of the simple magnet in an external field / M.P. Kozlovskii, I.V. Pylyuk, O.O. Prytula // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 749–760. — Бібліогр.: 20 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1210492025-06-03T16:27:05Z Behaviour of the order parameter of the simple magnet in an external field Поведінка параметра порядку простого магнетика в зовнішньому полі Kozlovskii, M.P. Pylyuk, I.V. Prytula, O.O. The effect of a homogeneous external field on the three-dimensional uniaxial magnet behaviour near the critical point is investigated within the framework of the nonperturbative collective variables method using the ρ⁴ model. The research is carried out for the low-temperature region. The analytic explicit expressions for the free energy, average spin moment and susceptibility are obtained for weak and strong fields in comparison with the field value belonging to the pseudocritical line. The calculations are performed on the microscopic level without any adjusting parameters. It is established that the long-wave fluctuations of the order parameter play a crucial role in forming a crossover between the temperature-dependence and fielddependence critical behaviour of the system. В рамках непертурбативного методу колективних змінних на основі моделі ρ⁴ досліджено вплив зовнішнього поля на критичну поведінку тривимірного одновісного магнетика. Дослідження проведені для низькотемпературної області. Отримані аналітичні вирази для вільної енергії, середнього спінового моменту і сприйнятливості у випадку слабких і сильних полів по відношенню до величини поля, що задовільняє рівняння псевдокритичної лінії. Розрахунки проведені на мікроскопічному рівні без застосування допоміжних параметрів. Встановлено, що вирішальну роль у формуванні переходу між температурозалежною і залежною від поля критичною поведінкою системи відіграють довгохвильові флуктуації параметра порядку. 2005 Article Behaviour of the order parameter of the simple magnet in an external field / M.P. Kozlovskii, I.V. Pylyuk, O.O. Prytula // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 749–760. — Бібліогр.: 20 назв. — англ. 1607-324X PACS: 05.50.+q, 05.70.Ce, 64.60.Fr, 75.10.Hk DOI:10.5488/CMP.8.4.749 https://nasplib.isofts.kiev.ua/handle/123456789/121049 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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The effect of a homogeneous external field on the three-dimensional uniaxial
magnet behaviour near the critical point is investigated within the framework
of the nonperturbative collective variables method using the ρ⁴ model.
The research is carried out for the low-temperature region. The analytic
explicit expressions for the free energy, average spin moment and susceptibility
are obtained for weak and strong fields in comparison with the field
value belonging to the pseudocritical line. The calculations are performed
on the microscopic level without any adjusting parameters. It is established
that the long-wave fluctuations of the order parameter play a crucial role
in forming a crossover between the temperature-dependence and fielddependence
critical behaviour of the system. |
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Article |
| author |
Kozlovskii, M.P. Pylyuk, I.V. Prytula, O.O. |
| spellingShingle |
Kozlovskii, M.P. Pylyuk, I.V. Prytula, O.O. Behaviour of the order parameter of the simple magnet in an external field Condensed Matter Physics |
| author_facet |
Kozlovskii, M.P. Pylyuk, I.V. Prytula, O.O. |
| author_sort |
Kozlovskii, M.P. |
| title |
Behaviour of the order parameter of the simple magnet in an external field |
| title_short |
Behaviour of the order parameter of the simple magnet in an external field |
| title_full |
Behaviour of the order parameter of the simple magnet in an external field |
| title_fullStr |
Behaviour of the order parameter of the simple magnet in an external field |
| title_full_unstemmed |
Behaviour of the order parameter of the simple magnet in an external field |
| title_sort |
behaviour of the order parameter of the simple magnet in an external field |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121049 |
| citation_txt |
Behaviour of the order parameter of the simple magnet in an external field / M.P. Kozlovskii, I.V. Pylyuk, O.O. Prytula // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 749–760. — Бібліогр.: 20 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kozlovskiimp behaviouroftheorderparameterofthesimplemagnetinanexternalfield AT pylyukiv behaviouroftheorderparameterofthesimplemagnetinanexternalfield AT prytulaoo behaviouroftheorderparameterofthesimplemagnetinanexternalfield AT kozlovskiimp povedínkaparametraporâdkuprostogomagnetikavzovníšnʹomupolí AT pylyukiv povedínkaparametraporâdkuprostogomagnetikavzovníšnʹomupolí AT prytulaoo povedínkaparametraporâdkuprostogomagnetikavzovníšnʹomupolí |
| first_indexed |
2025-12-02T08:40:51Z |
| last_indexed |
2025-12-02T08:40:51Z |
| _version_ |
1850385211585462272 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 749–760
Behaviour of the order parameter of the
simple magnet in an external field
M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received July 18, 2005, in final form October 31, 2005
The effect of a homogeneous external field on the three-dimensional uniax-
ial magnet behaviour near the critical point is investigated within the frame-
work of the nonperturbative collective variables method using the ρ
4 mod-
el. The research is carried out for the low-temperature region. The analytic
explicit expressions for the free energy, average spin moment and suscep-
tibility are obtained for weak and strong fields in comparison with the field
value belonging to the pseudocritical line. The calculations are performed
on the microscopic level without any adjusting parameters. It is established
that the long-wave fluctuations of the order parameter play a crucial role
in forming a crossover between the temperature-dependence and field-
dependence critical behaviour of the system.
Key words: critical point, order parameter, Ising model
PACS: 05.50.+q, 05.70.Ce, 64.60.Fr, 75.10.Hk
1. Introduction
We propose an approach for describing the critical behaviour of a three-dimen-
sional (3D) uniaxial magnet. Despite the variety of investigations, the problem has
not been solved exactly so far [1]. Another reason of studying this model is its
possible application to the study of nonmagnetic systems, such as binary alloys,
simple fluids, micellar systems and so on. The second order phase transitions in
the systems belonging to the 3D Ising universality class are also expected in high-
energy physics [2]. Most investigations are devoted to the calculations of universal
characteristics of the system, particularly, critical exponents and amplitude ratios
of thermodynamic functions.
The description of the system taking into account the effect of the external field
appears to be a more complicated problem. It is well known that the presence of
the field causes the smearing of second order phase transition. In the vicinity of the
critical point, the singularities of some thermodynamic functions transform into the
c© M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula 749
M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula
maxima at fixed values of the field and temperature. These values correspond to the
case when the effects of the field and temperature on the critical behaviour of the
system are equivalent. Therefore, the simple series expansions in the scaling variable
are not valid in this region, since this variable is of the order of unity [3]. In this
region, the relations of thermal properties of the spin models to the properties of
the clusters of spins in geometrical terms [4] are of great interest.
Early investigations regarding the 3D Ising model in the external field are re-
ported in [5]. It is performed along the critical “isoterm” (T = Tc) using the series
expansion technique. In [6], the description of the system is carried out employi-
ng the transfer matrix method. The asymptotic form of the low-temperature free
energy for this model was obtained. However, this description is valid only for low
temperatures. The results presented in [7] were obtained using the quantum field
theory and renormalization group (RG) technique. All calculations are based on the
perturbative expansion at fixed d = 3 dimension. The equation of state is deter-
mined numerically using the parametric representation. The main difficulty in using
this approach is to extrapolate the field theory results from T > Tc to T < Tc region
(Tc is the critical temperature). To solve this problem, it is necessary to proceed
with analytical continuation in the complex τ -plane (τ is the reduced temperature
(τ = (T −Tc)/Tc)). Consequently, the calculations of thermodynamic functions and
their ratios of amplitude are complex. In order to determine the equation of state
in parametric representation, one should employ some adjusting parameters. This is
another disadvantage of this approach. The equation of state in parametric represen-
tation can be also obtained using the high-temperature expansion results [8]. There
are other investigations devoted to this problem, that are performed by numerical
methods [9–12].
In this article, the description of the 3D Ising-like magnet near the critical point
in the external field by the nonperturbative collective variables (CV) method [13,14]
is presented for the case of T < Tc. Using the transition from the spin variables to
the collective variables, which play the role of the modes of spin density oscillations,
one can calculate both universal and nonuniversal characteristics of the system. Par-
ticularly, we determine the explicit expressions for free energy, average spin moment
and susceptibility as functions of the external field, which is introduced in the Hamil-
tonian from the outset. These calculations are based on the non-Gaussian quartic
measure density (ρ4 model). Since, the investigations are preformed on the micro-
scopic level and the interaction potential contains some microscopic parameters,
one can investigate the thermodynamic characteristics of the system as functions of
these parameters.
2. Basic relations
We consider the simplest one-component spin model on the simple cubic lattice
with period c in a homogeneous external field h. For calculations, we use the following
750
Behaviour of the order parameter
approximation of Fourier transform
Φ(k) =
{
Φ(0)(1 − 2b2k2), k ∈ B0,
Φ0 = Φ(0)Φ̄, k ∈ B\B0 ,
(1)
where Φ(0) = 8πA (b/c)3, and regions B and B0 are defined as
B =
{
k = (kx, ky, kz)|ki = −π
c
+
2π
c
ni
Ni
; ni = 1, 2, . . . , Ni, i = x, y, z
}
, (2)
B0 =
{
k = (kx, ky, kz)|ki = − π
c0
+
2π
c0
ni
N0i
; ni = 1, 2, . . . , N0i, i = x, y, z
}
. (3)
The quantities A and b are the microscopic parameters of the interaction potential
[13], Φ̄ is the small constant, N3
0i = N0, c0 = cs0, N0 = s−d
0 N , d = 3 is the dimension
of the space and N is total number of sites. The parameter s0 (s0 > 1) determines
the region of values k ∈ B0, where the parabolic approximation for Φ(k) is valid.
It should be noted that the interaction potential is an exponentially decreasing
function of the distance rjl between particles at sites j and l, Φ(rjl) = A exp(−rjl/b).
In the general form, the Fourier transform of this potential is determined as Φ(k) =
Φ(0)/(1 + b2k2)2. For small values of the wave vector, the parabolic approximation
is effective.
The starting point of these calculations is the partition function of the N0-
multiple integral with respect to CV [15]:
Z = Z0
√
2
N0−1
eã0N0
∫
(dρ)N0 exp
[
−a1
√
N0ρ0 −
1
2
∑
k∈B0
d(k)ρkρ−k
−a3
3!
N
−1/2
0
∑
k1,...,k3
ki∈B0
ρk1
. . . ρk3
δk1+···+k3
−a4
4!
N−1
0
∑
k1,...,k4
ki∈B0
ρk1
. . . ρk4
δk1+···+k4
]
, (4)
where
d(k) = ã2 − βΦ(0) + 2b2βΦ(0)k2, β = 1/kT. (5)
The appearance of the terms with odd powers of CV is caused by the presence
of the field. Coefficients Z0, ã0, ã2 and al are functions of the external field [15].
The detailed procedure of calculating (4) by the step-by-step integration method is
presented in [13,14]. It can be represented as RG transformation of the Wilson type
with renormalization parameter s (s > 1). After performing np + 1 iterations, as a
result, we obtain the partition function in the following form:
Z = Z0Q0Q1 . . . Qnp
jnp+1
[
Q(P (np))
]Nnp+1
Inp+1 . (6)
The partial partition functions are written as
Qn = [Q(P (n−1))Q(dn)]Nn .
751
M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula
Here jnp+1 =
√
2
Nnp+1−1
, and general expressions for Q(P (n−1)) and Q(dn) are given
in [16]. The quantity Inp+1 from (6) is presented as [15]
Inp+1 =
∫
(dρ)Nnp+1 exp
[
−ã
(np+1)
1 N
1/2
np+1ρ0 −
1
2
∑
k∈Bp
dnp+1(k)ρkρ−k
− 1
3!
a
(np+1)
3 N
−1/2
np+1
∑
k1,...k3
ki∈Bp
ρk1
. . . ρk3
δk1+···+k3
− 1
4!
a
(np+1)
4 N−1
np+1
∑
k1,...,k4
ki∈Bp
ρk1
. . . ρk4
δk1+···+k4
]
. (7)
The region Bp ≡ Bnp+1 has the form similar to (3) for the lattice with period
cnp+1 = c0s
np+1, where s is the RG parameter (s > 1). The coefficients in (7) are
expressed through initial coefficients using the recurrence relations (RR). Passing
on from quantities a
(n)
l to wn, rn, vn and un by relations
ã
(n)
1 = s−nwn, dn(0) = s−2nrn ,
a
(n)
3 = s−3nvn, a
(n)
4 = s−4nun , (8)
new quantities wn, rn, vn and un are defined by solutions of RR linearized near the
fixed point (w∗ = 0, r∗ = −f0βΦ(0), v∗ = 0, u∗ = ϕ0(βΦ(0))2). In the case of
T < Tc, they have the form
wn = −ch1h
′En
1 − ch2h
′T
(0)
13
(
ϕ
1/2
0 βΦ(0)
)−1
En
3 ,
rn = r∗ − c
(0)
k1 βΦ(0)τ1E
n
2 + ck2T
(0)
24 (ϕ
1/2
0 βΦ(0))−1En
4 ,
vn = −ch2h
′En
3 ,
un = u∗ − c
(0)
k1 (βΦ(0))2T
(0)
42 ϕ
1/2
0 τ1E
n
2 + ck2E
n
4 . (9)
Here h′ = βh is the reduced external field, τ1 = −τ , El are eigenvalues of the RG
transformation matrix (E1 = 20.977, E2 = 7.374, E3 = 1.838 and E4 = 0.397)
[15]. Other quantities are some coefficients, which do not depend on the field and
temperature [16]. The condition of small deviations of quantities wn, rn, vn and un
from the fixed point defines some value n = np, at which the system leaves the critical
regime region (the exit point). Taking into account the eigenvalues El, one can see,
that deviations are mainly formed in the first two equations of (9) by temperature
τ and field h variables. In order to obtain explicit dependences for thermodynamic
functions, we use the approximation, that the exit point np is only a function of one
of the variables h′ and τ . One chooses the variable, which has a stronger effect on
the critical behaviour than the other one. Hence, there are two cases: the weak field
region is determined by the equality
np = µτ = − ln τ̃1
ln E2
− 1, (10)
752
Behaviour of the order parameter
where the quantity µτ defines the exit point by the temperature value, and for the
strong field region we have
np = nh = − ln h̃
ln E1
− 1, (11)
where nh is the exit point controlled by the field. Here h̃ = h′/f0 and τ̃1 = c
(0)
k1 τ1/f0.
The equality µτ = nh corresponds to the case when the effects of the temperature
and the field on the critical behaviour of the system are equivalent. In the field-
temperature plane in double logarithmic scale of the plot, this case is represented
by the so-called pseudocritical line
h̃c = τ̃βδ
1 , (12)
where the quantities β and δ are the critical exponents.
3. Free energy of the system and equation of state
According to the formula (6), it is convenient to write down the free energy in
the form: [15]
Fe = F0 + FCR + FTR + FI . (13)
The term F0 corresponds to the contribution from the noninteracting spins (in the
case of Φ̄ = 0). It has the following form
F0 = −kTN
(
ln 2 + ln cosh h′ +
1
2
βΦ(0)Φ̄
)
. (14)
The term FCR is the result of RG transformations and represents the contribution
from the short-wave oscillation modes. The expression for FCR is written as
FCR = −kTN0
[
e0p − e1pτ̃1 + e2pτ̃
2
1 + e3ph̃
2
+(−F10 + F11τ̃1E
np+1
2 − F12τ̃
2
1 E
2(np+1)
2 )s−3(np+1)
]
. (15)
For the quantity FTR, we have
FTR = −kTN0s
−3(np+1)
{
fp1c − np ln s − fp11cτ̃1E
np+1
2 − fp12cτ̃
2
1 E
2(np+1)
2
}
. (16)
This is the free energy of the regime, which corresponds to the transition from short-
wave to long-wave oscillation modes of the order parameter. In expressions (15) and
(16), the quantities F1l, elp, fp1c and fp1lc are independent of the field [16].
The term from (13), which represents the contribution from the long-wave fluc-
tuations, has the form
FI = −kT ln Inp+1. (17)
753
M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula
The quantity Inp+1 is calculated using the series of substitutions of variables. The
first of them is
ρk = ηk + σh
√
Nδk.
Thus, we obtain
Inp+1 = exp [E0(σh)]
∫
(dη)Nnp+1 exp
[
A0
√
Nη0 −
1
2
∑
k∈Bnp+1
dh(k)ηkη−k
− 1
3!
bhN
−1/2
np+1
∑
k1,...,k3
ki∈Bnp+1
ηk1
. . . ηk3
δk1+···+k3
− 1
4!
ahN
−1
np+1
∑
k1,...,k4
ki∈Bnp+1
ηk1
. . . ηk4
δk1+···+k4
]
. (18)
Here
A0 = a1mh̃ − dnp+1(0)σh −
1
6
a
(np+1)
4 σ3
h
N
Nnp+1
,
dh(k) = dh(0) + 2βΦ(0)b2k2, dh(0) = dnp+1(0) +
1
2
a
(np+1)
4 σ2
h
N
Nnp+1
,
bh = σha
(np+1)
4
(
N
Nnp+1
)
1
2
, ah = a
(np+1)
4 (19)
and
E0(σh) = N
a1mσhh̃ − 1
2
dnp+1(0)σ2
h −
a
(np+1)
4
24
σ4
h
N
Nnp+1
. (20)
The quantity σh can be found employing the condition ∂E0(σh)/∂σh = 0. Performing
the substitution of the variable
σh = σ0s
−
np+1
2 , (21)
we arrive at the cubic equation
[
a1mh̃E
np+1
1 − rnp+1σ0 −
1
6
unp+1σ
3
0s
3
0
]
s−
5
2
(np+1) = 0. (22)
The solution of (22) is chosen using the condition of free energy minimization. It
also nullifies the quantity A0 from (19). The quadratic term in the expression of
the exponent in (18) becomes positive and dominating in comparison with other
terms for all k 6= 0. Thus, we can perform the integration in (18) with respect to the
variables ηk except the variable η0. The next step in the calculations lies in returning
to the variable ρ0 by means of
η0 = ρ0 − σh
√
N.
754
Behaviour of the order parameter
The average value of the variable ρ0 plays the role of the order parameter of the
system. Therefore, performing the integration with respect to this variable , we
get the main contribution to the total free energy and average spin moment. The
integration is carried out performing the substitution ρ0 = ρh
√
N and using the
steepest descent method. As a result, the quantity Inp+1 assumes the following form:
Inp+1 =
∏
k∈Bnp+1
k 6=0
(
π
dh(k)
)
1
2 √
N
∫
dρh exp [NE0] , (23)
where
E0 = a1mE
np+1
1 s−
5
2
(np+1)h̃ρh −
1
2
rnp+1s
−2(np+1)ρ2
h −
1
4!
sd
0unp+1s
−(np+1)ρ4
h . (24)
Using the condition ∂E0/∂ρh = 0, we find the root of the cubic equation satisfying
the condition of the free energy minimization in the form
ρh = σ0hs
−
np+1
2 . (25)
Thus, the quantity Inp+1 can be written as
Inp+1 =
∏
k∈Bnp+1
k 6=0
(
π
dh(k)
)
1
2
exp [NE0(σ0h)] , (26)
where
E0(σ0h) = s−3(np+1)
[
a1mE
np+1
1 h̃σ0h −
1
2
rnp+1σ
2
0h −
1
4!
sd
0unp+1σ
4
0h
]
. (27)
As one can see, this quantity is the function of the variable ρ0, whose average value
is the order parameter as it was mentioned above. Thus, the expression (27) is the
microscopic analog of the Landau free energy as well as the relation (25) is the
equation of state.
Taking the logarithm, transiting to the spherical Brillouin zone and integrating
with respect to k in the expression (26), we arrive at the formula for free energy of
long-wave fluctuations:
FI = −kTN
{[
(np + 1) ln s − 1
2
I ′
0 +
1
2
ln π
]
s−3
0 + E0(σ0h)
}
s−3(np+1). (28)
The quantity I ′
0 has the form
I ′
0 = ln(D′
0 + D′
1) −
2
3
+ 2
D′
0
D′
1
− 2
(
D′
0
D′
1
)
3
2
arctan
(
D′
1
D′
0
)
1
2
,
and coefficients D′
0 and D′
1 are defined as
D′
0 = rnp+1 +
1
2
s3
0unp+1σ
2
0h, D′
1 = 2βΦ(0)s−2
0
(
πb
c
)2
.
755
M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula
Collecting the contributions from all regimes of fluctuations according to (13), we
can now write down complete expressions for the free energy of the system in case
of strong and weak fields. Taking into account that in the case of np = µτ (see (10))
s−3(µτ+1) = τ̃ 3ν
1 and τ̃1E
µτ+1
2 = 1, the total free energy for the weak field region takes
on the form
Fe = −kTN
[
ln cosh h′ + l0 + l1µeτ̃
3ν
1 + l11µeh̃τ̃
ν
2
1 + l2h̃
2 − l3τ̃1 + l4τ̃
2
1
]
, (29)
where
l0 = ln 2 +
1
2
βcΦ(0)Φ̄ + s−3
0 e0p ,
l1µe = l
(0)
1µe −
1
2
rµτ+1σ
2
0h −
1
24
uµτ+1s
3
0σ
4
0h ,
l11µe = a1mσ0h , l2 = s−3
0 e3p ,
l3 = s−3
0 e1p −
1
2
βcΦ(0)Φ̄f0/c
(0)
k1 , l4 = s−3
0 e2p +
1
2
βcΦ(0)Φ̄f 2
0 /(c
(0)
k1 )2. (30)
Here βc is the inverse critical temperature. The quantity l
(0)
1µe is obtained by the
summation of coefficients proportional to τ̃ 3ν
1 from different contributions.
ρ
Figure 1. The contribution to the order parameter from the regime of long-wave
fluctuations as a function of the field for τ = 10−5.
According to the relation (11), we have s−3(nh+1) = h̃
6
5 and Enh+1
2 = h̃−
1
βδ . Thus,
the free energy of the strong field region is written as
Fe,h = −kTN
[
ln cosh h′ + l0 + l
(−)
1e h̃
6
5 − l11eτ̃1h̃
6
5
−
1
p0
+l12eτ̃
2
1 h̃
6
5
−
2
p0 + l2h̃
2 − l3τ̃1 + l4τ̃
2
1
]
. (31)
Here p0 = βδ, and the coefficients satisfy the following relations:
l
(−)
1e = E0,h + (ln s − 1
2
I ′
0 +
1
2
ln π + fp1c − F10)s
−3
0 ,
l11e = −s−3
0 (F11 − fp11c), l12e = −s−3
0 (F12 + fp12c). (32)
756
Behaviour of the order parameter
The main contribution to the order parameter (25) for weak fields h′ < h′
c takes on
the form
ρh = σ0hτ̃
ν
2
1 (33)
and for strong fields h′ > h′
c
ρh = σ0hh̃
1
5 , (34)
respectively. The quantity σ0h is the root of the cubic equation
a1mh̃E
np+1
1 − rnp+1σ0h −
1
6
unp+1σ
3
0hs
3
0 = 0 (35)
derived from (27). The dependences (33) and (34) are shown in figure 1. As one can
see, they coincide in the point h′ = h′
c (h′
c = f0h̃c). Tending to the pseudocritical line
region, the coefficient σ0h becomes essentially dependent on the field. In the point
h′ = h′
c, this dependence is the most substantial. Such a dependence is related to
the presence of the field in terms of the cubic equation (35). The quantity σ0h en-
sures a crossover between the temperature-dependence and field-dependence critical
behaviour. With the field further increasing, the dependence on the field is reduced,
and σ0h in the case of h′ � h′
c becomes practically independent of the field.
The expressions (29) and (31) allow us to get other thermodynamic characte-
ristics of the system. Particularly, the total average spin moment can be obtained
using the well-known relation σ = −(N−1 · dF/dh)T . In the case of the small fields,
the explicit expression for order parameter has the form
σ(−)
e = tanh h′ + 2l2f
−2
0 h′ + l11µef
−1
0 τ̃
ν
2
1 . (36)
The dependence of the coefficient l11µe on the field and temperature is responsible
for the crossover between two types of the critical behaviour and for the divergence
of the second order derivatives of the free energy in the critical point. When h̃ > h̃c,
the total order parameter can be written as follows
σ
(−)
e,h = tanh h′ + 2l2f
−2
0 h′ + σ0ehh
′
6
5 +
6
5
l
(−)
1e f
−
6
5
0 h′
1
5
−σ1ehτ̃1h
′
1
5
−
1
p0 + σ2ehτ̃
2
1 h
′
1
5
−
2
p0 . (37)
The coefficients in expressions (36), (37) and in the following expressions for the
susceptibility can be obtained through the coefficients (30) and (32) by differentia-
ting the free energy of the system with respect to the field variable for the case of
h̃ < h̃c and h̃ > h̃c, respectively.
In the case of h̃ < h̃c, the susceptibility is defined as
χ(−)
e =
1
kT
[1 − tanh2 h′ + 2l2f
−2
0 + f−1
0
∂l11µe
∂h
τ̃
ν
2
1 ]. (38)
When h̃ > h̃c, we obtain
χ
(−)
e,h =
1
kT
[
1 − tanh2 h′ + 2l2f
−2
0 + χ0ehh
′
6
5 +
12
5
σ0ehh
′
1
5
+χ
(−)
1ehh
′−
4
5 − χ2ehτ̃1h
′−
4
5
−
1
p0 + χ3ehτ̃
2
1 h
′−
4
5
−
2
p0
]
. (39)
757
M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula
Due to the approximation that exit point np is one-variable function, there is
some disagreement between the quantities σ(−)
e , σ
(−)
e,h as well as between χ(−)
e and
χ
(−)
e,h in the vicinity of the pseudocritical line (for example, χ(−)
e = 2.01 · 105 and
χ
(−)
e,h = 2.02 · 105 at τ = 10−5, h′ = h′
c). Nevertheless, the region of disagreement
is quite narrow, since the long-wave fluctuations play the major role in the critical
phenomena.
4. Conclusions
The description of the effect of the external magnetic field on the 3D Ising-like
magnet near the critical point using the CV method in the low-temperature region
is presented. We get the explicit analytic expressions for the free energy, order pa-
rameter and susceptibility as functions of the field and temperature. For variable ρ0,
which plays the role of the order parameter in the CV method, the field dependence
of the whole range of fields and temperatures in the vicinity of the critical point is
obtained. In the proposed approach, the crossover between critical behaviour con-
trolled by field or temperature variable is ensured mainly by the contribution from
the long-wave fluctuations. Therefore, this dependence is the main contribution to
the total equation of state. Since the calculations are carried out on the microscop-
ic level, we can obtain the dependences of the nonuniversal characteristics on the
microscopic parameters of the system (a lattice constant and parameters of the in-
teraction potential). Calculations are performed within the framework of the quartic
measure density, which allows one to obtain the qualitative description of the sys-
tem behaviour. For more accurate estimations, it is necessary to use the ρ6 model
[17–19]. The calculations can be also extended to the classical n-vector magnetic
model [20].
758
Behaviour of the order parameter
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759
M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula
Поведінка параметра порядку простого магнетика
в зовнішньому полі
М.П.Козловський, І.В.Пилюк, О.О.Притула
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 18 липня 2005 р., в остаточному вигляді –
31 жовтня 2005 р.
В рамках непертурбативного методу колективних змінних на основі
моделі ρ
4 досліджено вплив зовнішнього поля на критичну пове-
дінку тривимірного одновісного магнетика. Дослідження проведені
для низькотемпературної області. Отримані аналітичні вирази для
вільної енергії, середнього спінового моменту і сприйнятливості
у випадку слабких і сильних полів по відношенню до величини
поля, що задовільняє рівняння псевдокритичної лінії. Розрахунки
проведені на мікроскопічному рівні без застосування допоміжних
параметрів. Встановлено, що вирішальну роль у формуванні пе-
реходу між температурозалежною і залежною від поля критичною
поведінкою системи відіграють довгохвильові флуктуації параметра
порядку.
Ключові слова: критична точка, параметр порядку, модель Ізинга
PACS: 05.50.+q, 05.70.Ce, 64.60.Fr, 75.10.Hk
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