The critical region thermodynamics of some statistical models
We consider the exactly solvable Ashkin-Teller model and the hard square model, in which both scaling law hypothesis and the universality one are violated. The critical properties of these models have been investigated. The reasons for the violation of the scaling law and universality hypotheses...
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2005
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| Цитувати: | The critical region thermodynamics of some statistical models / E.D. Soldatova, O.M. Galdina // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 793–800. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1210542017-06-14T03:04:42Z The critical region thermodynamics of some statistical models Soldatova, E.D. Galdina, O.M. We consider the exactly solvable Ashkin-Teller model and the hard square model, in which both scaling law hypothesis and the universality one are violated. The critical properties of these models have been investigated. The reasons for the violation of the scaling law and universality hypotheses are clarified. Неабиякий інтерес для сучасної теорії критичних явищ становлять послідовні точно розв’язувані моделі статистичної механіки, в яких порушуються гіпотези подібності й універсальності. До них відносять модель Ешкіна-Теллера і модель жорстких квадратів. В даній роботі досліджено їх критичні властивості. Пояснюються причини порушення гіпотез подібності й універсальності. 2005 Article The critical region thermodynamics of some statistical models / E.D. Soldatova, O.M. Galdina // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 793–800. — Бібліогр.: 10 назв. — англ. 1607-324X PACS: 64.60.Fr DOI:10.5488/CMP.8.4.793 http://dspace.nbuv.gov.ua/handle/123456789/121054 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
We consider the exactly solvable Ashkin-Teller model and the hard square
model, in which both scaling law hypothesis and the universality one are
violated. The critical properties of these models have been investigated.
The reasons for the violation of the scaling law and universality hypotheses
are clarified. |
| format |
Article |
| author |
Soldatova, E.D. Galdina, O.M. |
| spellingShingle |
Soldatova, E.D. Galdina, O.M. The critical region thermodynamics of some statistical models Condensed Matter Physics |
| author_facet |
Soldatova, E.D. Galdina, O.M. |
| author_sort |
Soldatova, E.D. |
| title |
The critical region thermodynamics of some statistical models |
| title_short |
The critical region thermodynamics of some statistical models |
| title_full |
The critical region thermodynamics of some statistical models |
| title_fullStr |
The critical region thermodynamics of some statistical models |
| title_full_unstemmed |
The critical region thermodynamics of some statistical models |
| title_sort |
critical region thermodynamics of some statistical models |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121054 |
| citation_txt |
The critical region thermodynamics of some statistical models / E.D. Soldatova, O.M. Galdina // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 793–800. — Бібліогр.: 10 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT soldatovaed thecriticalregionthermodynamicsofsomestatisticalmodels AT galdinaom thecriticalregionthermodynamicsofsomestatisticalmodels AT soldatovaed criticalregionthermodynamicsofsomestatisticalmodels AT galdinaom criticalregionthermodynamicsofsomestatisticalmodels |
| first_indexed |
2025-07-08T19:06:48Z |
| last_indexed |
2025-07-08T19:06:48Z |
| _version_ |
1837106842061242368 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 793–800
The critical region thermodynamics of
some statistical models
E.D.Soldatova, O.M.Galdina
Dnipropetrovs’k National University
Received July 18, 2005, in final form October 31, 2005
We consider the exactly solvable Ashkin-Teller model and the hard square
model, in which both scaling law hypothesis and the universality one are
violated. The critical properties of these models have been investigated.
The reasons for the violation of the scaling law and universality hypotheses
are clarified.
Key words: stability coefficients, violation of the scaling law and the
universality hypotheses
PACS: 64.60.Fr
1. Preface
The successes of the modern theory of critical phenomena are connected with the
fundamental conclusions regarding the nature of the critical state and the existence
of the class of systems, for which both the scaling law hypothesis and the universality
one are fulfilled. However, some theses of this theory have not been strictly proved
so far. Besides, a special form of Hamiltonian is used. From this point of view the
existence of real systems and consistent models contradicting the above-mentioned
hypotheses are the problem of great importance. The 6-vertex Lieb model and the
hard square model, in which the scaling law hypothesis is violated, as well as Baxter
model and Ashkin-Teller one, in which the universality hypothesis is violated, are
the models of this kind. The main aim of this paper is to derive thermodynamic
properties of Ashkin-Teller model and the hard square model in the vicinity of
critical point. We analyze the behaviour of the whole set of stability characteristics
(the adiabatic and isodynamic parameters) in the critical region and determine the
critical behaviour types for these models based on the thermodynamic method for
the one-component system critical state. The method is based on the constructive
critical state definition and on the condition of critical state stability. It leads to
the existence of four types of critical behaviour [1,2]. We argue the reasons for the
c© E.D.Soldatova, O.M.Galdina 793
E.D.Soldatova, O.M.Galdina
violation of the scaling law and the universality hypotheses in Ashkin-Teller model
as well as in the hard square model.
Let us consider the basic theses of thermodynamic method.
2. The thermodynamic method for the critical state
The basic parameters characterizing a thermodynamic system from the stand-
point of its stability are the adiabatic parameters (the AP’s)
(
∂T
∂S
)
x
,
(
∂T
∂x
)
S
,
(
∂X
∂x
)
S
, and the isodynamic parameters (the IP’s)
(
∂T
∂S
)
X
,
(
∂T
∂x
)
X
,
(
∂X
∂x
)
T
.
The parameters
(
∂T
∂S
)
x
and
(
∂X
∂x
)
S
are called the adiabatic stability coefficients
(the ASC’s), and the parameters
(
∂T
∂S
)
X
and
(
∂X
∂x
)
T
are called the isodynamic
stability coefficients (the ISC’s) [3,4]. They are related to the fluctuations of the ex-
ternal system parameters (the first and the second Gibbs lemmas), which infinitely
increase near the critical point. One can consider the volume V , the magnetization
M and the electric polarization P as the generalized thermodynamic variables (the
external parameters) x. The conjugated thermodynamic forces are the pressure P ,
the electric field strength E and the magnetic field strength H.
The most complete critical state definition, which includes both the properties of
heterogeneous and homogeneous system (in the subcritical and supercritical region,
correspondingly) can be written in the following form [1,2]:
dT =
(
∂T
∂S
)
x
dS +
(
∂T
∂x
)
S
dx = 0
dX =
(
∂X
∂S
)
x
dS +
(
∂X
∂x
)
S
dx = 0
,
(
∂X
∂T
)
c
= −
dS
dx
= Kc , (1)
where Kc is the critical slope of a phase equilibrium curve. For the existence of
non-trivial solutions of (1) the condition
(
∂T
∂S
)
x
(
∂T
∂x
)
S(
∂T
∂x
)
S
(
∂X
∂x
)
S
= D =
(
∂T
∂S
)
x
(
∂X
∂x
)
S
−
(
∂T
∂x
)2
S
= 0 (2)
has to be fulfilled all over the spinodal.
It coincides with the well-known critical state condition D = 0, where D is the
stability determinant [3,4].
The definition (1) describes the critical state by means of the AP’s. The critical
slope Kc is the solution of a set of homogeneous linear equations (1)
−
dS
dx
= Kc =
[
sign
(
∂T
∂x
)
S
]
(
(
∂X
∂x
)
S
(
∂T
∂S
)−1
x
)1/2
. (3)
794
The critical region thermodynamics of some statistical models
It is the fundamental characteristic of the critical state and expressed by the ASC’s.
This definition combined with the critical state stability conditions [1,2] leads to
a variety of critical state manifestations, namely, to the existence of four alternative
types of critical behaviour of thermodynamic systems:
1.
(
∂T
∂S
)
x
6= 0,
(
∂X
∂x
)
S
6= 0 ⇒
(
∂T
∂x
)
S
6= 0, Kc 6= {0,∞}
2.
(
∂T
∂S
)
x
6= 0,
(
∂X
∂x
)
S
= 0 ⇒
(
∂T
∂x
)
S
= 0, Kc = 0
3.
(
∂T
∂S
)
x
= 0,
(
∂X
∂x
)
S
6= 0 ⇒
(
∂T
∂x
)
S
= 0, Kc = ∞
4.
(
∂T
∂S
)
x
= 0,
(
∂X
∂x
)
S
= 0 ⇒
(
∂T
∂x
)
S
= 0, Kc is not defined by (1).
(4)
The critical behaviour type of a specific physical system is defined by the value
of one ASC and Kc. The behaviour of the whole set of stability characteristics
(the AP’s and IP’s) can be determined for each critical type. The fourth type of
critical behaviour is the most interesting and the most “fluctuating” one. In this
case both the ASC’s tend to zero (and, hence, the rest of stability characteristics
tend to zero). As a consequence the critical slope is not determined by set (1). In
this case it is necessary to consider the differential equations of higher orders. Then
the solution is realized through several possibilities [1,2]. The case of two or even
three phase equilibrium curves converging at the critical point is of special interest.
As we demonstrated in [5], the critical point of the ferroelectric Lieb model is of this
kind.
3. The critical properties of Ashkin-Teller model
J.Ashkin and E.Teller [6] suggested their model as a generalization of Ising model
for the four-component system. Each lattice site in this model is occupied by an atom
of one of the four kinds: A, B, C or D. The interaction energy of two neighbours is
equal to ε0 for AA, BB, CC, DD; ε1 for AB, CD; ε2 for AC, BD and ε3 for AD,
BC.
The model can be expressed by the Ising spins. Each site i is associated with
two spins: si and σi. Let (si, σi) = (+, +), (+,−), (−, +), (−,−) if site i is occupied
by the atom A, B, C, D, correspondingly. Then the interaction energy for the edge
(i, j) has the form
ε(i, j) = −Jsisj − J ′σiσj − J4siσisjσj − J0 , (5)
−J = (ε0 + ε1 − ε2 − ε3)/4,−J ′ = (ε0 + ε2 − ε3 − ε1)/4,
−J4 = (ε0 + ε3 − ε1 − ε2)/4,−J0 = (ε0 + ε1 + ε2 + ε3)/4. (6)
The partition function is defined by the expression
ZAT =
∑
s
∑
σ
exp
−
∑
(i,j)
ε(i, j)/kT
, (7)
795
E.D.Soldatova, O.M.Galdina
where k is Boltzmann constant and T is the temperature. The sum in the exponent
runs over all the curves (i, j), which join the neighbouring sites of the lattice; the
outer sums are taken over the values of all the spins s1, s2, s3, . . . and σ1, σ2, σ3, . . .
The dimensionless coefficients of interaction and the Boltzmann weighting coef-
ficients are defined by
K = J/kT, K ′ = J ′/kT,
K4 = J4/kT, K0 = J0/kT, (8)
ω0 = exp (K + K ′ + K4 + K0) , ω1 = exp (K − K ′ − K4 + K0) ,
ω2 = exp (−K + K ′ − K4 + K0) , ω3 = exp (−K − K ′ + K4 + K0) . (9)
This consideration is valid for any lattice, not necessarily the plane one. Usually,
the square lattice with N sites is analyzed. As it follows from (5), Ashkin-Teller
model can be presented as two Ising models on a square lattice (s-model and σ-
model) connected by means of the four-spin interactions.
The model is also similar to the eight-vertex model [7] in the absence of external
field with the Hamiltonian
H = −
M
∑
i=1
N
∑
j=1
{Jvσijσi,j+1 + Jhσijσi+1,j
+ Jσi,j+1σi+1,j + J ′σijσi+1,j+1 + J ′′σijσi,j+1σi+1,j} ,
in which Jv = Jh = 0. However, the lattice geometry is different in these two cases.
The spins of the eight-vertex model are located at different sites, whereas the spins
of both kinds si and σi are located at the same site i in Ashkin-Teller model.
The critical exponents for Ashkin-Teller model can be written in the form:
α =
2 − 2y
3 − 2y
, βm =
2 − y
24 − 16y
, γm =
14 − 7y
12 − 8y
, δm = 15;
βe =
1
12 − 8y
, γe =
7 − 4y
6 − 4y
, δe = 15 − 8y, (10)
where y is the interaction parameter which is related to the Baxter model interaction
parameter [7] in the following way y = 2µ/π, 0 < µ < 2π/3.
Therefore, there are two sets of critical exponents: the “magnetic” exponents,
corresponding to the field −H
∑
σi, and the “electric” exponents, corresponding to
the field −E
∑
σisi. Here we use indices m and e for the “magnetic” and “electric”
exponents. So, βm is the critical exponent for the order parameter 〈σ1〉 and βe is the
critical exponent for the order parameter 〈σ1s1〉.
Both Ashkin-Teller model and Baxter model disobey the universality principle,
since their critical exponents depend on the interaction parameter y continuously.
Thus, the isodynamic stability coefficients for the “magnetic” Ashkin-Teller mod-
el have the following asymptotic form
(
∂T
∂S
)
H
∼ (−t)
2 − 2y
3 − 2y ,
(
∂H
∂M
)
T
∼ (−t)
14 − 7y
12 − 8y ,
796
The critical region thermodynamics of some statistical models
where t = (T − Tc)/Tc . In the absence of the external field (this case is the subject
of consideration here) the isodynamic and the adiabatic parameters coincide. Hence,
the asymptotic expressions for adiabatic parameters are as follows:
(
∂T
∂S
)
M
∼ (−t)
2 − 2y
3 − 2y ,
(
∂H
∂M
)
S
∼ (−t)
14 − 7y
12 − 8y ,
(
∂T
∂M
)
S
∼ (−t)
26 − 17y
24 − 16y .
Let us analyze the ASC’s behaviour. When 1 < y < 4/3, that corresponds to
π/2 < µ < 2π/3, the critical exponent α becomes negative, the exponent γ is
positive, i.e. the second type of the critical behaviour with the critical slope value
Kc = 0 is realized. All the inverse susceptibilities (the adiabatic and isothermic) as
well as the parameters
(
∂T
∂M
)
S
and
(
∂T
∂M
)
H
tend to zero at the critical point,
and the thermic stability coefficients
(
∂T
∂S
)
M
and
(
∂T
∂S
)
H
are finite. At y = 1 the
critical exponents are as follows: α = 0, β = 1/8, γ = 7/4. This coincides with the
2D Ising model results [9] and corresponds to the fourth type of critical behaviour
with Kc = 0. At 0 6 y < 1 (0 6 µ < π/2) the exponent α becomes positive and
α < γ – the fourth type of critical behaviour with Kc = 0 is fulfilled. When y = 0,
we have α = 2/3, β = 1/12, γ = 7/6, i.e. all the critical exponents coincide with the
critical exponents of the three-spin model [10] realizing the fourth type of critical
behaviour with Kc = 0.
For the “electric” Ashkin-Teller model the isodynamic stability coefficients are
as follows:
(
∂T
∂S
)
E
∼ (−t)
2 − 2y
3 − 2y ,
(
∂E
∂P
)
T
∼ (−t)
7 − 4y
6 − 4y ,
and the adiabatic parameters are
(
∂T
∂S
)
P
∼ (−t)
2 − 2y
3 − 2y ,
(
∂E
∂P
)
S
∼ (−t)
7 − 4y
6 − 4y ,
(
∂T
∂P
)
S
∼ (−t)
13 − 8y
12 − 8y .
At the values of the interaction parameter 1 6 y < 4/3, which corresponds
to π/2 6 µ < 2π/3, the second type of critical behaviour is also fulfilled. When
0 6 y < 1 (0 6 µ < π/2), the critical exponent α is always less than the critical
exponent γ, and the fourth type of critical behaviour with critical slope Kc = 0 is
realized. The case y = 0, as above, corresponds to the three-spin model.
Consequently, the performed analysis shows, that the violation of the univer-
sality hypothesis in Ashkin-Teller model is concerned with the dependence of the
critical behaviour type on the interaction parameter y. The second or the fourth
type of critical behaviour is possible, which is characterized by the different order
of fluctuation growth at the critical point.
797
E.D.Soldatova, O.M.Galdina
4. The features of critical behaviour of the hard square model
One of the exactly solvable models of statistical physics is the hard hexagon
model [8]. It is the two-dimensional lattice model of hard molecules (the molecules,
which do not overlap). In this model the particles are situated at sites of the tri-
angular lattice in such a way, that any two of them are not allowed to be located
both at the same site or at neighbouring sites. Let us imagine that each particle is
situated at the centre of hexagon, which covers the six neighbouring edges of the
triangular lattice. Then all these hexagons do not overlap. That could explain the
name of this model.
Choosing the lattice with the edges of the right-angled triangular form, we obtain
a more general model, in which the nearest neighbour sites cannot be occupied
and the particles at the diagonals of square edges interact. Each spin σi can take
the values 1 or 0. This model is called a hard square model [8]. It is the limiting
case of the lattice gas with the interaction between the nearest neighbours. There
is a phase transition from the homogeneous liquid state at small activities z to
the inhomogeneous solid state at large activities. The partition function can be
expressed as
Z =
∑
σ
zσ1+···+σN
∏
(i,j)
(1 − σiσj),
where the product is carried out all over the edges (i, j) of the lattice, and the
summation is performed all over the values (0, 1) for all quantities σ1, . . . , σN .
The critical exponents in this models are as follows:
α′ = −
1
2
, α =
3
4
, β =
1
4
, (11)
i.e. the critical exponent for the heat capacity is different in the subcritical and
supercritical region α 6= α′. It means that the scaling law hypothesis is violated
in the model. We have already dealt with such a situation in Lieb model [5]. The
stability coefficients in the subcritical region are
(
∂T
∂S
)
V
∼ (−t)−1/2,
(
−
∂P
∂V
)
S
= const,
i.e.
(
∂T
∂S
)
V
= 0,
(
−
∂P
∂V
)
S
= 0
The critical slope is Kc 6= {0,∞}, so, the first type of critical behaviour is realized.
In the supercritical region
(
∂T
∂S
)
V
∼ (−t)3/4,
(
−
∂P
∂V
)
S
∼ (−t)3/4,
i.e.
(
∂T
∂S
)
V
6= 0,
(
−
∂P
∂V
)
S
6= 0,
798
The critical region thermodynamics of some statistical models
so, the fourth type of critical behaviour with the critical slope of the phase equilib-
rium curve Kc 6= {0,∞} is fulfilled.
These are the distinct types of critical behaviour in the sense of fluctuation na-
ture – the second type is characterized by the minimal order of the energy and
compressibility fluctuation growth. For the fourth type all the fluctuations are ex-
tremely large. The reason for the violation of the scaling law hypothesis in the hard
square models concerns this very fact.
5. Conclusions
The paper considers the thermodynamic stability of Ashkin-Teller model and
the hard square model using the method [1,2]. The asymptotic expressions for the
whole set of stability characteristics are determined and the reasons for the violation
of the scaling law and the universality hypotheses in given models are clarified. In
the hard square model, which can be applied to describing the critical behaviour
of a liquid, the first type of critical behaviour is realized in the subcritical region,
whereas the fourth type is fulfilled in the supercritical region. The violation of the
scaling law hypothesis can be explained just through the difference of the behaviour
types.
In Ashkin-Teller model which describes the critical behaviour of (anti)ferromag-
nets and (anti)ferroelectrics the second and fourth type of critical behaviour takes
place. The reason for the violation of the universality hypothesis is that each of the
mentioned types is related to the value of interaction parameter. It is interesting to
emphasize that in both models the violation of one hypothesis does not spoil another
one. Therefore, the capabilities of the thermodynamic method for a one-component
system critical state are illustrated using the above-mentioned models. The global
reasons are revealed for the violation of the scaling law and the universality hy-
potheses concerning the variety of the critical state manifestations.
References
1. Soldatova E.D. Thermodynamic Stability in Critical State Region. Thesis for a Doctor’s
degree (Physics and Mathematics). Kyiv, KSU, 1991 (in Russian).
2. Soldatova E.D., Cond. Matt. Phys., 1999, 2, No. 4(20), 603–616.
3. Semenchenko V.K. Selected Chapters of Theoretical Physics. Prosveschenie, Moscow,
1966.
4. Semenchenko V.K., Crystallography, 1964, 9, No. 5, 611–621 (in Russian).
5. Soldatova E.D., Galdina O.M., Ukr. J. Phys., 2004, 49, No. 11, 1122–1126.
6. Ashkin J., Teller E., Phys. Rev., 1943, 64, No. 5–6, 178–184.
7. Baxter R.J., Stud. Appl. Math, 1970, L, No. 1, 51–69.
8. Baxter R.J. Exactly Solvable Models in Statistical Mechanics. Academic Press, London,
1982.
9. Stanley H.E. Introduction to Phase Transitions and Critical Phenomena. Clarendon
Press, Oxford, 1971.
10. Baxter R.J., Enting I.G., J. Phys. A: Math. Gen., 1976, 9, L149–L151.
799
E.D.Soldatova, O.M.Galdina
Термодинаміка критичної області деяких
статистичних моделей
Є.Д.Солдатова, О.М.Галдіна
Дніпропетровський національний університет
Отримано 18 липня 2005 р., в остаточному вигляді –
31 жовтня 2005 р.
Неабиякий інтерес для сучасної теорії критичних явищ становлять
послідовні точно розв’язувані моделі статистичної механіки, в
яких порушуються гіпотези подібності й універсальності. До них
відносять модель Ешкіна-Теллера і модель жорстких квадратів.
В даній роботі досліджено їх критичні властивості. Пояснюються
причини порушення гіпотез подібності й універсальності.
Ключові слова: коефіцієнти стійкості, порушення гіпотез
подібності й універсальності
PACS: 64.60.Fr
800
|