The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field
The four-dimensional ferromagnetic Ising model in external magnetic field is simulated on the Creutz cellular automaton algorithm using finite-size lattices with linear dimension 4 ≤ L ≤ 8. The critical temperature value of infinite lattice, Tc χ (∞) = 6.680(1) obtained for h = 0 agrees well with...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1196942017-06-09T03:03:22Z The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field Ziya Merdan Cihan Kürkçü Mustafa K. Öztürk Низкотемпеpатуpный магнетизм The four-dimensional ferromagnetic Ising model in external magnetic field is simulated on the Creutz cellular automaton algorithm using finite-size lattices with linear dimension 4 ≤ L ≤ 8. The critical temperature value of infinite lattice, Tc χ (∞) = 6.680(1) obtained for h = 0 agrees well with the values Tc(∞) ≈ 6.68 obtained previously using different methods. Moreover, h = 0.00025 in our work also agrees with all the results obtained from h = 0 in the literature. However, there are no works for h ≠ 0 in the literature. The value of the field critical exponent (δ = 3.0136(3)) is in good agreement with δ = 3 which is obtained from scaling law of Widom. In spite of the finitesize scaling relations of |ML(t)| and χ L(t) for 0 ≤ h ≤ 0.001 are verified; however, in the cases of 0.0025 ≤ h ≤ 0.1 they are not verified. 2014 Article The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field / Ziya Merdan, Cihan Kürkçü, Mustafa K. Öztürk // Физика низких температур. — 2014. — Т. 40, № 12. — С. 1360-1365. — Бібліогр.: 31 назв. — англ. 0132-6414 PACS 05.50.+q, 64.60.Cn, 75.40.Cx, 75.40.Mg http://dspace.nbuv.gov.ua/handle/123456789/119694 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм |
| spellingShingle |
Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм Ziya Merdan Cihan Kürkçü Mustafa K. Öztürk The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field Физика низких температур |
| description |
The four-dimensional ferromagnetic Ising model in external magnetic field is simulated on the Creutz cellular
automaton algorithm using finite-size lattices with linear dimension 4 ≤ L ≤ 8. The critical temperature value of
infinite lattice, Tc
χ (∞) = 6.680(1) obtained for h = 0 agrees well with the values Tc(∞) ≈ 6.68 obtained previously
using different methods. Moreover, h = 0.00025 in our work also agrees with all the results obtained from h = 0
in the literature. However, there are no works for h ≠ 0 in the literature. The value of the field critical exponent
(δ = 3.0136(3)) is in good agreement with δ = 3 which is obtained from scaling law of Widom. In spite of the finitesize
scaling relations of |ML(t)| and χ L(t) for 0 ≤ h ≤ 0.001 are verified; however, in the cases of 0.0025 ≤ h ≤ 0.1
they are not verified. |
| format |
Article |
| author |
Ziya Merdan Cihan Kürkçü Mustafa K. Öztürk |
| author_facet |
Ziya Merdan Cihan Kürkçü Mustafa K. Öztürk |
| author_sort |
Ziya Merdan |
| title |
The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field |
| title_short |
The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field |
| title_full |
The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field |
| title_fullStr |
The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field |
| title_full_unstemmed |
The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field |
| title_sort |
finite-size scaling study of four-dimensional ising model in the presence of external magnetic field |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2014 |
| topic_facet |
Низкотемпеpатуpный магнетизм |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119694 |
| citation_txt |
The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field / Ziya Merdan, Cihan Kürkçü, Mustafa K. Öztürk // Физика низких температур. — 2014. — Т. 40, № 12. — С. 1360-1365. — Бібліогр.: 31 назв. — англ. |
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Физика низких температур |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 12, pp. 1360–1365
The finite-size scaling study of four-dimensional Ising
model in the presence of external magnetic field
Ziya Merdan1, Cihan Kürkçü2, and Mustafa K. Öztürk1
1Faculty of Arts and Sciences, Department of Physics, Gazi University, Ankara, Turkey
2Faculty of Arts and Sciences, Department of Physics, Ahi Evran University, Kirsehir, Turkey
E-mail: zmerdan1967@hotmail.com
Received May 12, 2014, revised May 27, 2014, published online October 22, 2014
The four-dimensional ferromagnetic Ising model in external magnetic field is simulated on the Creutz cellular
automaton algorithm using finite-size lattices with linear dimension 4 ≤ L ≤ 8. The critical temperature value of
infinite lattice, Tc
χ (∞) = 6.680(1) obtained for h = 0 agrees well with the values Tc(∞) ≈ 6.68 obtained previously
using different methods. Moreover, h = 0.00025 in our work also agrees with all the results obtained from h = 0
in the literature. However, there are no works for h ≠ 0 in the literature. The value of the field critical exponent
(δ = 3.0136(3)) is in good agreement with δ = 3 which is obtained from scaling law of Widom. In spite of the finite-
size scaling relations of |ML(t)| and χ L(t) for 0 ≤ h ≤ 0.001 are verified; however, in the cases of 0.0025 ≤ h ≤ 0.1
they are not verified.
PACS: 05.50.+q Lattice theory and statistics (Ising, Potts, etc.);
64.60.Cn Order-disorder transformations;
75.40.Cx Static properties (order parameter, static susceptibility, heat capacities, critical exponents,
etc.);
75.40.Mg Numerical simulation studies.
Keywords: Ising model, finite-size scaling, cellular automaton, external magnetic field.
1. Introduction
While the four-dimensional ferromagnetic Ising model
is not directly applicable to real magnetic systems, it is
useful to investigate the influence of dimensionality on
phase transitions [1]. In fact, in Euclidean quantum field
theory, the 4d ferromagnetic Ising model describes the
physical dimension. As the dimensionality and/or the lat-
tice size increases, the simulation of the ferromagnetic
Ising model by the conventional Monte Carlo method be-
comes impractical and faster algorithms are needed. The
Creutz cellular automaton algorithm [2] does not require
high-quality random numbers, it is an order of magnitude
faster than the conventional Monte Carlo method and
compared to the Q2R cellular automaton [3], it has the
advantage of fluctuating internal energy from which the
specific heat can be computed.
The question of the four-dimensional ferromagnetic
Ising model exact solution with an external magnetic field
is not known. In two dimensions, the solution of ferromag-
netic Ising model is investigated [4–7]. By considering
different approximate methods, such as approximate two-
dimensional ferromagnetic Ising model solutions are pre-
sented [8–12]. In addition, the four-dimensional ferromag-
netic Ising model solution is approximated by using Creutz
cellular automaton algorithm with nearest neighbor inter-
actions and near the critical region [13–17]. The algorithm
of approximating finite size behavior of ferromagnetic
Ising model is extended to higher dimension [13–24]. It is
established that the algorithm has been powerful in terms
of providing the values of static critical exponents near the
critical region in four and higher dimensions with nearest
neighbor interactions [13–24].
In this paper, we simulated the four-dimensional ferro-
magnetic Ising model in the presence of an external magnet-
ic field with the Creutz cellular automaton algorithm. The
value of the field critical exponent (δ) is obtained. The fi-
nite-size scaling relations for ( )LM t and ( )L tχ are verified
for 0 0.001h≤ ≤ , where ( )LM t is the absolute value of the
magnetization and ( )L tχ is the magnetic susceptibility.
The model is described in Sec. 2, the results are discus-
sed in Sec. 3 and a conclusion is given in Sec. 4.
© Ziya Merdan, Cihan Kürkçü, and Mustafa K. Öztürk, 2014
mailto:zmerdan1967@hotmail.com
The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field
2. Model
Five binary bits are associated with each site of the lat-
tice. The value for each site is determined from its value
and those of its nearest neighbors at the previous time step.
The updating rule, which defines a deterministic cellular
automaton, is as follows: Of the five binary bits on each
site, the first one is the Ising spin iB . Its value may be “0”
or “1”. The Ising spin energy in the presence of an external
magnetic field, IH , is described by the Hamiltonian of the
form
I i j i
ij i
H J S S h S
< > < >
= − −∑ ∑ (1)
taking into account of the interaction between the nearest
neighbors and also interaction of the spins iS with external
magnetic field h, directed “up” ( 1)iS = + . The spins affect-
ed by the field are directed “up” and not changed during
the simulation. Therefore, these spins play the role of the
magnetic field. In the Hamiltonian, 2 1i iS B= − and h is
the ratio of the number of “up” spins to the number of all
spins. The next three bits are for the momentum variable
conjugate to the spin (the demon). These three bits form an
integer which can take on the values within the interval
(0,7). The kinetic energy (in units of J ) associated with the
demon can take four times these integer values. The total
energy
I KH H H= + (2)
is conserved, where IH is the Ising spin energy and KH is
the kinetic energy of the lattice. For a given total energy
the system temperature T (in units of / BJ k where Bk is
the Boltzmann constant) is obtained from the average val-
ue of the kinetic energy. The fifth bit provides a checker-
board style updating, and so it allows the simulation of the
ferromagnetic Ising model on a cellular automaton. The
black sites of the checkerboard are updated and then their
color is changed into white. White sites are changed into
black without being updated. The updating rules for the
spin and the momentum variables are as follows: For a site
to be updated, its spin is flipped and the change in the Ising
spin energy (internal energy), IH , is calculated. If this en-
ergy change is transferable to or from the momentum vari-
able associated with this site, such that the total energy H
is conserved, then this change is done and the momentum
is appropriately changed. Otherwise the spin and the mo-
mentum are not changed.
As the initial configuration all the spins are taken to be
ordered (up or down). The initial kinetic energy is given to
the lattice via the first and the third bits of the momentum
variables in the white sites randomly, such that the value of
the initial kinetic energy for such a demon is 20 (in units of
J ), which is just the amount needed to flip a spin at its
initial configuration.
Simulations are carried out on simple hypercubic lattic-
es 4L of linear dimensions 4 8L≤ ≤ with periodic bounda-
ry conditions. The cellular automaton develops 59.6·10
( 4, 6, 8)L = sweeps for each run, with 7 runs for each total
energy.
3. Results and discussion
The temperature dependence of the order parameter and
the magnetic susceptibility for several values of h are illus-
trated in Fig. 1 for the lattice with L = 8. These functions
exhibit a similar behavior for a lattice of linear dimensions
L = 4 and 6. The temperature dependence of the order pa-
rameter is in an agreement with the expected behavior in
the presence of the field (Fig. 1(a)). As the external mag-
netic field is increased, the peaks of the magnetic suscepti-
bility decrease with respect to the peak values for 0h = .
For 0h = , the magnetic susceptibility peak is very sharp.
As the external magnetic field is increased, the temperature
dependence of the magnetic susceptibility becomes smooth
(Fig. 1(b)). The critical temperatures of the finite-size lat-
tices obtained from the magnetic susceptibility maxima
( )cT Lχ for 0 0.1h≤ ≤ are listed in Table 1.
Fig. 1. The temperature dependence of the order parameter and
the magnetic susceptibility for several values of h for L = 8.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 12 1361
Ziya Merdan, Cihan Kürkçü, and Mustafa K. Öztürk
Table 1. The maximum values and the critical temperatures of
the magnetic susceptibility for 0 0.1h≤ ≤
L cT χ χmax h
4 6.513(21) 0.912(20)
0
6 6.611(23) 2.340(34)
8 6.635(11) 4.467(32)
4 6.514(1) 0.908(12)
0.00025
6 6.629(3) 2.330(43)
8 6.640(9) 4.452(2)
4 6.515(1) 0.889(4)
0.00050
6 6.631(1) 2.325(11)
8 6.659(1) 4.440(9)
4 6.524(1) 0.865(1)
0.001
6 6.637(56) 2.321(7)
8 6.672(63) 4.363(16)
4 6.586(81) 0.862(9)
0.0025
6 6.680(5) 2.262(1)
8 6.759(32) 4.166(1)
4 6.592(12) 0.857(1)
0.0050
6 6.786(11) 2.000(2)
8 6.859(23) 3.882(3)
4 6.891(13) 0.817(5)
0.01
6 6.933(67) 1.617(3)
8 6.982(56) 3.260(54)
4 7.168(1) 0.703(21)
0.025
6 7.246(3) 1.097(31)
8 7.249(1) 2.811(18)
4 7.741(12) 0.473(25)
0.050 6 7.766(31) 0.642(42)
8 7.788(14) 1.906(12)
4 8.455(9) 0.271(1)
0.1
6 8.495(12) 0.377(4)
8 8.524(15) 1.024(12)
The dependence of the critical temperatures ( )cT Lχ ob-
tained from the magnetic susceptibility maxima of the fi-
nite-size lattices on linear dimension L is given by the fol-
lowing expression [13–17,25,26]:
2 1/6( ) ( ) logc cT T L L Lχ χ − −∞ − ∝ . (3)
The values of the infinite-lattice critical temperature
for the four-dimensional ferromagnetic Ising model,
6.680(1) ( ) 8.539(3)cT≤ ∞ ≤ are obtained from the straight
line fit of the magnetic susceptibility maxima for
0 0.1h≤ ≤ (Fig. 2). The value obtained of infinite lattice
critical temperature ( )cT χ ∞ = 6.680(1) for 0h = agrees well
with the value ( ) 6.68cT ∞ ≈ obtained previously using dif-
ferent methods [13–17,27–29]. However, there are no
works for 0h ≠ in the literature; moreover, 0.00025h = in
our work agrees with all the results obtained from 0h = in
the literature. In this work, cT are obtained for 0h ≠ and
given in Table 2. In our opinion, symmetry is conserved
for 0h = and 0.00025. That is why the results agree with
the values in the literature. However, broken symmetry
occurs for the values of h > 0.00025.
Table 2. The values of the infinite-lattice critical temperature
for 0 0.1h≤ ≤
cT χ Method
6.6802(2) [27] Series expansion
6.6803(1) [27] Dynamic Monte Carlo
6.680(1) [28,29] Cluster Monte Carlo
6.680, 6.6802 [13–17] Creutz cellular automaton
6.680(1) for h = 0, this work Creutz cellular automaton
6.680(3) for h = 0.00025, this work Creutz cellular automaton
6.691(28) for h = 0.00050, this work Creutz cellular automaton
6.723(15) for h = 0.001, this work Creutz cellular automaton
6.793(52) for h = 0.0025, this work Creutz cellular automaton
6.938(85) for h = 0.0050, this work Creutz cellular automaton
6.997(62) for h = 0.01, this work Creutz cellular automaton
7.284(45) for h = 0.025, this work Creutz cellular automaton
7.797(51) for h = 0.050, this work Creutz cellular automaton
8.539(3) for h = 0.1, this work Creutz cellular automaton
Fig. 2. The values of the infinite-lattice critical temperature for
the four-dimensional ferromagnetic Ising model with ν = 1/2,
( )cT χ ∞ = 6.680(1) for h = 0 (R2 = 0.995, marked ◊), ( )cT χ ∞ =
= 6.680(3) for h = 0.00025 (R2 = 0.946, □), ( )cT χ ∞ = 6.691(28)
for h = 0.00050 (R2 = 0.995, ∆), ( )cT χ ∞ = 6.723(15) for h =
= 0.001 (R2 = 0.999, ○), ( )cT χ ∞ = 6.793(52) for h = 0.0025 (R2 =
= 0.948, *), ( )cT χ ∞ = 6.938(85) for h = 0.0050 (R2 = 0.999, ),
( )cT χ ∞ = 6.997(62) for h = 0.01 (R2 = 0.898, ), ( )cT χ ∞ =
= 7.284(45) for h = 0.025 (R2 = 0.955, ▲), ( )cT χ ∞ = 7.797(51)
for h = 0.050 (R2 = 0.948, ●) and ( )cT χ ∞ = 8.539(3) for
h = 0.01 (R2 = 0.967, +), obtained by extrapolating temperatures
of the lattice with the linear dimension 4 ≤ L ≤ 8 as L → ∞ .
1362 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 12
The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field
For a lattice linear dimension L and very small h at
( )cT T L= , the order parameter is given by
1/ ( )( ) LM L h δ∝ , (4)
where ( )Lδ is the field critical exponent (Fig. 3). Scaling
law of Widom is following:
( 1)γ = β δ − , (5)
where 1, 1/ 2γ = β = and 3δ = for 4d = [30,31]. Since the
magnetic susceptibility in the presence of the external
magnetic field is smooth, the finite-size lattice critical tem-
peratures for the values of each h are obtained from the
locus of points of maximum slope in the magnetization
results as Monte Carlo calculations [8].
The log-log plots of ( )M L at ( , )cT T L h= versus h
for h in the interval 0.00025 0.1h≤ ≤ yields to 1/ ( )Lδ
(Fig. 4). The straight line which fits to the plot of ( )Lδ
against 1/ L results in the infinite-lattice critical exponents
3.0136(3)δ = (Fig. 5). The result for the ( )δ ∞ is compared
with 3δ = which is obtained from scaling law of Widom
[30,31].
Privman–Fisher hypothesis for the singular part of the
free-energy density ( ) ( , )S
Lf t h of a hypercubic finite system
dL with periodic boundary conditions is adapted for the
Ising model in 4d = dimensions, by proposing the finite-
size scaling function ( , )Y x y , correct to leading logarithms
as below:
( ) 4 2 1/6 3 1/4( , ) ( log , log ),
0, 0, ,
S
Lf t h L Y tL L hL L
t h L
−=
→ → → ∞
(6)
where ( ) /c ct T T T= − is the reduced temperature and h is
the external magnetic field [13]. From Eq. (6) the finite-
size scaling expressions for the magnetization ( , )LM t h ,
the magnetic susceptibility ( , )L t hχ can be derived as be-
low:
1 1/4 2 1/6 3 1/4( , ) log ( ) ( log , log ),L
L
f
M t h L L U tL L hL L
h
−∂
= − =
∂
(7)
2
2 1/2 2 1/6 3 1/4
2( , ) log ( ) ( log , log ),L
L
f
t h L L V tL L hL L
h
∂
χ = − =
∂
(8)
where ,U V are the corresponding finite-size scaling func-
tions. For 0h = they reduced to the following equations
[13]:
/ 1/4 2 1/6( ) log ( ) ( log )LM t L L U tL L−β ν= , (9)
/ 1/2 2 1/6( ) log ( ) ( log )L t L L V tL Lγ νχ = , (10)
where β , γ and ν are critical exponents for the magnetiza-
tion, the magnetic susceptibility and the correlation length
Fig. 3. The dependence of M against h for the lattices with the
linear dimension 4 ≤ L ≤ 8 (Tc = 6.6802(2)).
Fig. 4. The log-log plots of M(L,Tc(L,h)) against h with the slope
giving the value of 1/δ = 0.187 for L = 4, 6 and 8 at Tc =
= 6.6802(2).
Fig. 5. The plot of δ(L) against 1/L. The extrapolation of the fit
lines to 1/L → ∞ gives δ = 3.0136(3).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 12 1363
Ziya Merdan, Cihan Kürkçü, and Mustafa K. Öztürk
of the infinite lattice, respectively. The relations for
0 0.1h≤ ≤ can be tested by simulations directly. The fi-
nite-size scaling plots for |ML(t)| and χ L(t) are given in
Figs. 6, 7, respectively. The overlap of the plots of the
scaled quantities for different L (Figs. 6(a),(b) and 7(a),(b))
verifies the finite-size scaling relations given in Eqs. (9)
and (10) at Tc = 6.6802(2). Owing to the overlap of the
plots of the scaled quantities for different L (Figs. 6(c) and
7(c)) doesn’t verify the finite-size scaling relations given in
Eqs. (9) and (10) at Tc = 6.6802(2).
4. Conclusions
The four-dimensional ferromagnetic Ising model in ex-
ternal magnetic field is simulated on the Creutz cellular
automaton algorithm by using the finite-size lattices with
the linear dimensions L = 4, 6, and 8. In our work, the crit-
ical temperature value of infinite lattice for h = 0 is in
agreement with the other simulation results. Moreover, h =
= 0.00025 in our work also agrees with all the results ob-
tained from h = 0 in the other simulation results in litera-
ture. The values h > 0.00025 do not agree with Tc values in
Fig. 6. Finite-size scaling plot of |ML| with β /ν = 1 and
Tc = 6.6802(2) for h = 0 (a), 0.00050 (b), 0.0050 (c).
Fig. 7. Finite-size scaling plot of χ L with γ /ν = 2 and
Tc = 6.6802(2) for h = 0 (a), 0.00050 (b), 0.0050 (c).
1364 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 12
The finite-size scaling study of four-dimensional Ising model in the presence of external magnetic field
the literature because broken symmetry occurs. In this
study, the value of the field critical exponent (δ =
= 3.0136(3)) is satisfied by scaling law of Widom. Al-
though the finite-size scaling relations of |ML(t)| and χL(t)
for 0 0.001h≤ ≤ are verified; however, in the cases of
0.0025 0.1h≤ ≤ are not verified. The values of h
(0.0025 0.1)h≤ ≤ aren’t verified because of broken sym-
metry. Finally, we note that the inconsistency in tempera-
ture, the absolute value of the magnetization and χL(t) is
the magnetic susceptibility are the result of broken sym-
metry and this affects the phase transition in finite systems.
The computer used to run the model is an Intel(R) Core
Duo CPU at 1830 MHz. Since the CPU time was invested
11200 h for all the simulations, the case of L ≥ 10 will be
considered later.
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 12 1365
1. Introduction
2. Model
3. Results and discussion
4. Conclusions
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/NLD (Gebruik deze instellingen om Adobe PDF-bestanden te maken die moeten worden gecontroleerd of die moeten voldoen aan PDF/A-1b, een ISO-standaard voor de langetermijnopslag \(archivering\) van elektronische documenten. Zie voor meer informatie over het maken van PDF/A-compatibele PDF-documenten de Acrobat-gebruikershandleiding. PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/ENU (Use these settings to create Adobe PDF documents that are to be checked or must conform to PDF/A-1b, an ISO standard for the long-term preservation \(archival\) of electronic documents. For more information on creating PDF/A compliant PDF documents, please refer to the Acrobat User Guide. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
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>> setdistillerparams
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/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|