Microscopic analogue of the Landau free energy for three-dimensional Ising-like systems
A microscopic analogue of the Landau free energy for a three-dimensional one-component spin system is found below the critical temperature as a result of direct calculations. The obtained explicit expressions make it possible to analyse the dependence of coefficients of the analogue on temperatur...
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Інститут фізики конденсованих систем НАН України
2003
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nasplib_isofts_kiev_ua-123456789-1207122025-06-03T16:30:16Z Microscopic analogue of the Landau free energy for three-dimensional Ising-like systems Мікроскопічний аналог вільної енергії Ландау для тривимірних ізінгоподібних систем Kozlovskii, M.P. Pylyuk, I.V. Prytula, O.O. A microscopic analogue of the Landau free energy for a three-dimensional one-component spin system is found below the critical temperature as a result of direct calculations. The obtained explicit expressions make it possible to analyse the dependence of coefficients of the analogue on temperature and microscopic parameters of the system. In contrast to the case for the Landau theory, the temperature dependence of these coefficients is nonanalytic. The quantities determining the coefficients in the expression for a microscopic analogue of the Landau free energy as well as the temperature-dependence curves for the order parameter of the system are given for different values of the effective radius of the exponentially decreasing interaction potential. Мікроскопічний аналог вільної енергії Ландау для тривимірної однокомпонентної спінової системи знайдено нижче критичної температури в результаті прямих розрахунків. Отримані явні вирази дозволяють дослідити залежність коефіцієнтів аналогу від температури та мікроскопічних параметрів системи. На відміну від теорії Ландау температурна залежність цих коефіцієнтів є неаналітичною. Величини, що визначають коефіцієнти у виразі для мікроскопічного аналогу вільної енергії Ландау, а також температурно залежні криві для параметра порядку системи, подані для різних значень радіуса ефективної дії експоненціально спадного потенціалу взаємодії. 2003 Article Microscopic analogue of the Landau free energy for three-dimensional Ising-like systems / M.P. Kozlovskii, I.V. Pylyuk, O.O. Prytula // Condensed Matter Physics. — 2003. — Т. 6, № 2(34). — С. 197-204. — Бібліогр.: 17 назв. — англ. 1607-324X PACS: 05.50.+q, 64.60.Fr, 75.10.Hk DOI:10.5488/CMP.6.2.197 https://nasplib.isofts.kiev.ua/handle/123456789/120712 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
A microscopic analogue of the Landau free energy for a three-dimensional
one-component spin system is found below the critical temperature as a
result of direct calculations. The obtained explicit expressions make it possible
to analyse the dependence of coefficients of the analogue on temperature
and microscopic parameters of the system. In contrast to the case
for the Landau theory, the temperature dependence of these coefficients
is nonanalytic. The quantities determining the coefficients in the expression
for a microscopic analogue of the Landau free energy as well as the
temperature-dependence curves for the order parameter of the system are
given for different values of the effective radius of the exponentially decreasing
interaction potential. |
| format |
Article |
| author |
Kozlovskii, M.P. Pylyuk, I.V. Prytula, O.O. |
| spellingShingle |
Kozlovskii, M.P. Pylyuk, I.V. Prytula, O.O. Microscopic analogue of the Landau free energy for three-dimensional Ising-like systems Condensed Matter Physics |
| author_facet |
Kozlovskii, M.P. Pylyuk, I.V. Prytula, O.O. |
| author_sort |
Kozlovskii, M.P. |
| title |
Microscopic analogue of the Landau free energy for three-dimensional Ising-like systems |
| title_short |
Microscopic analogue of the Landau free energy for three-dimensional Ising-like systems |
| title_full |
Microscopic analogue of the Landau free energy for three-dimensional Ising-like systems |
| title_fullStr |
Microscopic analogue of the Landau free energy for three-dimensional Ising-like systems |
| title_full_unstemmed |
Microscopic analogue of the Landau free energy for three-dimensional Ising-like systems |
| title_sort |
microscopic analogue of the landau free energy for three-dimensional ising-like systems |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120712 |
| citation_txt |
Microscopic analogue of the Landau
free energy for three-dimensional
Ising-like systems / M.P. Kozlovskii, I.V. Pylyuk, O.O. Prytula // Condensed Matter Physics. — 2003. — Т. 6, № 2(34). — С. 197-204. — Бібліогр.: 17 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kozlovskiimp microscopicanalogueofthelandaufreeenergyforthreedimensionalisinglikesystems AT pylyukiv microscopicanalogueofthelandaufreeenergyforthreedimensionalisinglikesystems AT prytulaoo microscopicanalogueofthelandaufreeenergyforthreedimensionalisinglikesystems AT kozlovskiimp míkroskopíčniianalogvílʹnoíenergíílandaudlâtrivimírnihízíngopodíbnihsistem AT pylyukiv míkroskopíčniianalogvílʹnoíenergíílandaudlâtrivimírnihízíngopodíbnihsistem AT prytulaoo míkroskopíčniianalogvílʹnoíenergíílandaudlâtrivimírnihízíngopodíbnihsistem |
| first_indexed |
2025-12-02T06:45:28Z |
| last_indexed |
2025-12-02T06:45:28Z |
| _version_ |
1850377952136527872 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 2(34), pp. 197–204
Microscopic analogue of the Landau
free energy for three-dimensional
Ising-like systems
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 October 14, 2002, in final form February 12, 2003
A microscopic analogue of the Landau free energy for a three-dimensional
one-component spin system is found below the critical temperature as a
result of direct calculations. The obtained explicit expressions make it pos-
sible to analyse the dependence of coefficients of the analogue on temper-
ature and microscopic parameters of the system. In contrast to the case
for the Landau theory, the temperature dependence of these coefficients
is nonanalytic. The quantities determining the coefficients in the expres-
sion for a microscopic analogue of the Landau free energy as well as the
temperature-dependence curves for the order parameter of the system are
given for different values of the effective radius of the exponentially de-
creasing interaction potential.
Key words: Ising-like system, collective variables, Landau free energy,
order parameter
PACS: 05.50.+q, 64.60.Fr, 75.10.Hk
1. Object of investigation and method
We consider a three-dimensional (3D) Ising-like system on a simple cubic lattice
with period c. The interaction potential is an exponentially decreasing function
Φ(rjl) = A exp(−rjl/b). Here, A is a constant, rjl is the interparticle distance, and
b is the radius of effective interaction. The approximation for the Fourier transform
of the interaction potential is taken in the form [1]
Φ̃(k) =
{
Φ̃(0)(1 − 2b2k2), k 6 B′,
0, B′ < k 6 B,
(1)
where B is the boundary of Brillouin half-zone (B = π/c), B ′ = (b
√
2)−1, Φ̃(0) =
8πA(b/c)3.
c© M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula 197
M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula
A microscopic analogue of the Landau free energy for 3D Ising-like systems is
calculated in the present paper using the collective variables (CV) method (see [1]).
The CV method allows us to calculate the partition function of the system and
to obtain not only the universal quantities (critical exponents) but also the nonuni-
versal characteristics. The methods existing at present make it possible to calculate
universal quantities to a quite high degree of accuracy (see, for example, [2–5]). The
advantage of the proposed method is the possibility of deriving analytic expressions
for the phase transition temperature and the amplitudes of thermodynamic char-
acteristics as functions of microscopic parameters of the initial system (the lattice
constant and parameters of the interaction potential), which makes this method
useful in describing the phase transitions in a wide class of 3D systems.
The term collective variables is applied to a special class of variables specific for
each individual physical system. The set of CV contains variables associated with
order parameters. For this reason, the phase space of CV is the most natural one
to use for describing a phase transition. For magnetic systems, the CV ρk are the
variables associated with modes of spin moment density oscillations, while the order
parameter is associated with the variable ρ0, in which the subscript “0” corresponds
to the peak of the Fourier transform of the interaction potential. An important
factor in describing the system behaviour near the phase transition temperature Tc
by the CV method is the use of non-Gaussian measure densities. A non-Gaussian
density of measure at a zero external field is represented as an exponential function of
the CV, the argument of which contains, along with the quadratic term, even higher
powers of the variable with the corresponding coupling constants. The simplest non-
Gaussian measure density is the quartic one (the ρ4 model) with the second and the
fourth powers of the variable in the exponent. The sextic measure density (the ρ6
model) includes the sixth power of the variable in addition to the second and the
fourth powers, etc. The quartic approximaton allows us to describe all qualitative
aspects of the second-order phase transition, while the sextic approximaton ensures
a more adequate quantitative description of the critical properties of a spin system.
This is confirmed by calculations as well as by the analysis of the behaviour of the
coefficients in the initial expression for partition function and the critical exponent
of the correlation length for the sequence of ρ4, ρ6, ρ8, and ρ10 models [1,6–8] as well
as by the calculation and comparison of thermodynamic functions for the models ρ4
and ρ6 [9] and by comparison of the results of our calculations with other available
data (see, for example, [9,10]).
In this paper, the critical behaviour of a 3D Ising-like system is studied on the
basis of the higher non-Gaussian approximation (the ρ6 model) taking into account
the correction-to-scaling terms.
The starting point of the problem statement in the CV method is the Hamil-
tonian of a 3D Ising-like system. After passing to the CV set, the Jacobian of the
transition from the spin variables to the CV is calculated to obtain a partition func-
tion functional similar to the Ginzburg-Landau functional. The partition function of
the spin system is integrated over the layers of the CV phase space. The correspond-
ing renormalization group (RG) transformation can be related to the Wilson type.
198
Microscopic analogue of the Landau free energy. . .
Although the CV method like the Wilson approach exploits the RG ideas, it is based
on the use of a non-Gaussian density of measure. The main feature is the integration
of short-wave spin density oscillation modes, which is generally done without using
perturbation theory. For this purpose, we divide the phase space of the CV ρk into
layers with the division parameter s. In each n-th layer (corresponding to the region
of wave vectors Bn+1 < k 6 Bn, Bn+1 = Bn/s, s > 1), the Fourier transform of the
potential Φ̃(k) is replaced by its average value (the arithmetic mean in the given
case). To simplify the presentation, we assume that the correction for the potential
averaging is zero, although it can be taken into account if necessary [1]. Including
this correction leads to a nonzero value of the critical exponent η characterizing the
behaviour of the pair correlation function for T = Tc. As a result of step-by-step cal-
culation of partition function, the number of integration variables in the expression
for this quantity decreases gradually. The partition function is then represented as
a product of partial partition functions Qn of separate layers and the integral of the
“smoothed” effective measure density W (n+1)
6 (ρ):
Z = 2N2(Nn+1−1)/2Q0Q1 · · ·Qn[Q(Pn)]Nn+1
∫
W(n+1)
6 (ρ) (dρ)Nn+1 . (2)
Here Nn+1 = N ′s−3(n+1), N ′ = Ns−3
0 , s0 = B/B′ = π
√
2b/c. The sextic density of
measure of the (n + 1)th block structure W (n+1)
6 (ρ) has the form
W(n+1)
6 (ρ) = exp
−1
2
∑
k6Bn+1
dn+1(k)ρkρ−k
−
3
∑
l=2
a
(n+1)
2l
(2l)!N l−1
n+1
∑
k1,...,k2l6Bn+1
ρk1
· · · ρk2l
δk1+···+k2l
, (3)
where δk1+···+k2l
is the Kronecker symbol, Bn+1 = B′s−(n+1), dn+1(k) = a
(n+1)
2 −
βΦ̃(k), β = 1/(kT ). The coefficients a
(n+1)
2l are renormalized values of the initial
coefficients a′
2l after integration over n + 1 layers of the phase space of CV. The
quantities a′
2l are determined by special functions [9,10] and are functions of s0, i.e.,
of the ratio b/c.
The basic idea of the calculation of explicit expressions for free energy and other
thermodynamic functions of the system near Tc on a microscopic level (τ < τ ∗ ∼
10−2, τ = (T − Tc)/Tc) lies in the separate inclusion of contributions from short-
wave and long-wave modes of spin moment density oscillations [1,11]. The short-
wave modes are characterized by the presence of RG symmetry and are described
by a non-Gaussian measure density. Here, the RG method is used (see, for example,
[12]). These modes are responsible for the formation of critical exponents and for
renormalization of the coefficient of the distribution describing the long-wave modes.
The way in which the contribution from long-wave modes of oscillations to the free
energy of the system is taken into account differs qualitatively from the method
of calculating the short-wave part of the partition function. The calculation of this
199
M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula
contribution is based on using the Gaussian density of measure as the basis density.
We have developed a direct method of calculation with the results obtained by taking
into account the short-wave modes as initial parameters.
Calculating separately the contributions to the free energy from short- and long-
wave modes of spin density oscillations, we can obtain a complete expression for the
free energy of the system. Detailed calculations of the contributions to the free ener-
gy of the system from short- and long-wave modes and the coefficients of complete
expressions for the entropy, internal energy, specific heat are presented in [13,14]. A
calculation technique for the first correction to scaling is elaborated in the course of
determining the thermodynamic functions of the system in the ρ6 model approxima-
tion. It is shown that each of the leading critical amplitudes and confluent correction
amplitudes can be represented as a product of a universal factor not depending on
the microscopic parameters of the system and a nonuniversal factor depending on
these parameters.
The suggested approach makes it possible to investigate the dependences of ther-
modynamic characteristics of a 3D Ising-like system on its microscopic parameters
[15].
2. Microscopic analogue of the Landau free energy and order
parameter of a 3D Ising-like system
The role of the order parameter for the system under investigation is played by
the average spin moment. It is associated with the existence of a nonzero value ρ̄0
below the phase transition temperature, for which the integrand of the expression
for the long-wave part of the partition function
Zµτ+1 = e−βF ′
µτ +1
∫
exp
[
β
√
Nρ0h + B̃ρ2
0 −
G
N
ρ4
0 −
D
N2
ρ6
0
]
dρ0 (4)
attains its extremum value. Here β is the inverse temperature, h is determined by the
value of the constant external magnetic field H introduced in our analysis (h = µBH,
µB being the Bohr magneton). The expression for −βF ′
µτ +1 corresponding to the
contribution to the free energy of the system from CV ρk with the values of wave
vectors k → 0 (but not equal to zero) as well as the coefficients of the expressions
B̃ = B̃(0)|τ |2νβΦ̃(0)(1 + B̃(1)|τ |∆1),
G = G(0)|τ |ν(βΦ̃(0))2(1 + G(1)|τ |∆1),
D = D(0)(βΦ̃(0))3(1 + D(1)|τ |∆1) (5)
are given in [15,16]. Here ν and ∆1 are the critical exponent of the correlation length
and the exponent of the first correction to scaling, respectively. Carrying out in (4)
the substitution of the variable
ρ0 =
√
Nρ, (6)
we obtain
Zµτ+1 = e−βF ′
µτ +1
√
N
∫
e−NE0(ρ)dρ, (7)
200
Microscopic analogue of the Landau free energy. . .
Table 1. Values of quantities determining the coefficients in the expression for a
microscopic analogue of the Landau free energy.
b bI bII bIII c 2c
s = 2.0000
B̃(0) 1.0106 0.9530 0.9305 0.7258 0.7149
B̃(1) −0.2733 −0.3959 −0.4420 −0.8188 −0.8375
G(0) 0.0550 0.0857 0.1010 1.9382 15.3880
G(1) −0.8919 −1.2918 −1.4423 −2.6720 −2.7330
D(0) 0.0009 0.0023 0.0033 1.5614 99.9318
D(1) −0.6952 −0.9377 −1.0470 −1.9396 −1.9839
s = 2.7349
B̃(0) 0.9417 0.8888 0.8683 0.6865 0.6768
B̃(1) −0.4451 −0.5124 −0.5377 −0.7445 −0.7550
G(0) 0.0690 0.1074 0.1267 2.4478 19.4434
G(1) −1.1718 −1.3491 −1.4157 −1.9601 −1.9876
D(0) 0.0012 0.0031 0.0044 2.0825 133.281
D(1) −0.8853 −1.0193 −1.0696 −1.4809 −1.5017
s = 3.0000
B̃(0) 0.9115 0.8610 0.8415 0.6697 0.6605
B̃(1) −0.4755 −0.5321 −0.5533 −0.7261 −0.7348
G(0) 0.0732 0.1141 0.1346 2.6087 20.7264
G(1) −1.1967 −1.3392 −1.3926 −1.8275 −1.8495
D(0) 0.0013 0.0033 0.0047 2.2185 141.986
D(1) −0.9113 −1.0199 −1.0606 −1.3918 −1.4085
and the evaluation of the order parameter is reduced to determining the extremum
point ρ̄ of the expression
E0(ρ) = Dρ6 + Gρ4 − B̃ρ2 − βhρ. (8)
The value of ρ̄ coincides with the average value of ρ corresponding to the equilibrium
value of the order parameter [1,9,10]. The expression for E0(ρ) defines the fraction
of free energy associated with the order parameter. It corresponds to a microscopic
analogue of the Landau free energy. The quantity Zµτ+1 will be expressed in terms of
E0(ρ̄) (coinciding in form with the expansion of the free energy into a power series
in the order parameter) by using the steepest descent method for evaluating the
integral (7) (see [16]).
The expression (8) was derived by successive elimination of “insignificant” vari-
ables ρk with k 6= 0, which allowed us to calculate the coefficients of E0(ρ) (see
table 1). Numerical values in table 1 are given for some values of the effective ra-
dius b of the potential and optimal values of the RG parameter s [9]. The value of
b = bI = c/(2
√
3) corresponds to the interaction between the nearest neighbours,
201
M.P.Kozlovskii, I.V.Pylyuk, O.O.Prytula
b = bII = 0.3379c corresponds to the interaction between the nearest and next-
nearest neighbours, and b = bIII = 0.3584c corresponds to the nearest, next-nearest,
and third neighbours [17]. At these values of b and at small values of the wave vec-
tors k, the parabolic approximation of the Fourier transform of the exponentially
decreasing interaction potential corresponds to the analogous approximation of the
Fourier transform for the interaction potentials of the above-mentioned neighbours.
For s = s∗ = 2.7349, the average value of the coefficient in the term with the second
power of the variable in the expression for the effective measure density is equal to
zero at a fixed point (in the ρ4 model, this corresponds to s∗ = 3.5862). Thus, there
is no need to postulate a temperature dependence of the coefficients in equation (8)
(as in the case of the Landau expansion) since the analytic form of their depen-
dence on temperature and microscopic parameters of the system has been obtained
as a result of direct calculations. Unlike the Landau theory case, the temperature
dependence of these coefficients is nonanalytic (see (5)).
Let us go over to direct calculations of the average spin moment. The point ρ̄
can be determined from the condition for the extremum ∂E0(ρ)/∂ρ = 0 or
6Dρ̄5 + 4Gρ̄3 − 2B̃ρ̄ − h
kT
= 0. (9)
For h = 0, we obtain the biquadratic equation
6Dρ̄4 + 4Gρ̄2 − 2B̃ = 0. (10)
Solving this equation and separating temperature explicitly, we arrive at the follow-
ing formula for the average spin moment 〈σ〉 = ρ̄:
〈σ〉 = 〈σ〉(0) |τ |β(1 + 〈σ〉(1) |τ |∆1). (11)
Here, β = ν/2 is the critical exponent of the average spin moment, and the coeffi-
cients 〈σ〉(l) are given in [15,16]. The curves describing the dependence of 〈σ〉 on τ
for various values of b are shown in figure 1 for s = 3.
3. Conclusions
The partition function of a 3D Ising-like system is integrated over the layers of
the CV phase space using the sextic measure density. Integration is performed over
all variables except for the variable ρ0 connected with the order parameter. Partition
function is reduced to a single integral. The exponent of the integrand contains the
energy E0(ρ) (see (7), (8)), which corresponds to a microscopic analogue of the
Landau free energy. The expression for E0(ρ) can be regarded as the part of free
energy associated with the order parameter.
The coefficients of a microscopic analogue of the Landau free energy as functions
of temperature and microscopic parameters of the system are obtained on the basis
of analytic calculations. In contrast to the Landau theory, the dependence of these
coefficients on the temperature is nonanalytic. Numerical values of the quantities
202
Microscopic analogue of the Landau free energy. . .
Figure 1. The temperature dependence of the average spin moment of the system
in the ρ6 model approximation for various values of the effective radius b of the
potential: bI = c/(2
√
3), bII = 0.3379c, bIII = 0.3584c, c, and 2c.
determining the coefficients of a microscopic analogue of the Landau free energy are
given for various values of the interaction potential range.
Due to the factor N in the exponent, the integrand in (7) possesses a sharp
maximum at the point ρ̄, which corresponds to the equilibrium value of the order
parameter. The average spin moment playing the role of the order parameter for the
investigated system is found.
The proposed method for computing a one-component spin system may be gen-
eralized to the case of a system with an n-component order parameter.
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Мікроскопічний аналог вільної енергії Ландау для
тривимірних ізінгоподібних систем
М.П.Козловський, І.В.Пилюк, О.О.Притула
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 14 жовтня 2002 р., в остаточному вигляді –
12 лютого 2003 р.
Мікроскопічний аналог вільної енергії Ландау для тривимірної одно-
компонентної спінової системи знайдено нижче критичної темпера-
тури в результаті прямих розрахунків. Отримані явні вирази дозво-
ляють дослідити залежність коефіцієнтів аналогу від температури
та мікроскопічних параметрів системи. На відміну від теорії Ландау
температурна залежність цих коефіцієнтів є неаналітичною. Величи-
ни, що визначають коефіцієнти у виразі для мікроскопічного аналогу
вільної енергії Ландау, а також температурно залежні криві для па-
раметра порядку системи, подані для різних значень радіуса ефек-
тивної дії експоненціально спадного потенціалу взаємодії.
Ключові слова: ізінгоподібна система, колективні змінні, вільна
енергія Ландау, параметр порядку
PACS: 05.50.+q, 64.60.Fr, 75.10.Hk
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