Finite Size Effect in the Quasi-One-Dimensional Ising Model

Journal of the Physical Society of Japan
Vol. 71, No. 11, November, 2002, pp. 2591–2594
#2002 The Physical Society of Japan
Finite Size Effect in the Quasi-One-Dimensional Ising Model
Kyu Won LEE
Natural Science Research Institute, Jeonju University, Jeonju 560-759, Korea
(Received April 3, 2002)
We have studied the finite size effect in the quasi-one-dimensional Ising model by using a Monte Carlo
simulation. A marked finite size effect was found with decreasing interchain interaction. Aside from the
well-known dimensional crossover from 3D to 1D, a dimensional crossover from 3D to 2D at a crossover
size Lcð Þ 1=ð ln Þ was revealed as an origin of the finite size effect, where is the ratio of the
interchain interaction to the intrachain one.
KEYWORDS: finite size effect, dimensional crossover, Monte Carlo, quasi-1D Ising model
DOI: 10.1143/JPSJ.71.2591
The quasi-one-dimensional (quasi-1D) Ising model is a
3D Ising model with a variable intechain interaction and is
described by the model Hamiltonian
H ¼ J X X
SjðrÞ Sjðr 0 Þ J X X
SjðrÞ Sjþ1ðrÞ;
j
r;r 0
ð1Þ
where the index j denotes the planes, and r and r0 denote a
spin coordinate in a plane. ¼ 1 corresponds to the 3D Ising
model. In the limit ! 0, the system becomes a collection
of noninteracting 1D Ising chains. One of the interesting
issues related to the quasi-1D model is a dimensional
crossover from 3D to 1D when is decreased.
Recently, many progressive experiments for the quasilow-dimensional
magnets were reported. 1–6) So-called
molecule-based magnets provide a good testing ground for
a dimensional crossover, in which the interchain interaction
can be easily controlled by changing the interchain distance.
In contrast to the common expectation for the quasi-1D
Heisenberg model, the magnetic ordering temperatures in
the molecule-based magnets were observed to be independent
of interchain distance for large interchain separations,
which was attributed to the interchain dipolar interaction. 1)
1D arrays of transition metal chains built by an atomic
engineering technique showed a ferromagnetic ordering at
finite temperature, which was attributed to an anisotropy
energy. 2) The magnetic nanodot 3) and the molecular
magnet 4) renewed the importance of the finite size effect,
outside of the finite size scaling region. In this work, we
have carried out a Monte Carlo study in order to elucidate
the crucial role of the system size in the quasi-1D Ising
model, which is related to the anisotropic growth of the
correlation length.
An asymptotic form of the critical temperature in the
quasi-1D Ising model was suggested as 7,8)
Tcð Þ
J
2 ln 1
ln ln 1 þ
j
r
1
: ð2Þ
The asymptotic form of eq. (2) was verified over a wide
range of in a previous Monte Carlo study. 9) However, the
anisotropic nature of the quasi-1D system seems not to have
been fully taken into account in previous works because they
considered only a single correlation length. In the anisotropic
system, the correlation length will grow anisotropi-
E-mail: rhdns@jeonju.ac.kr
2591
cally, so critical behavior may differ in different
directions. 10) As shown in a previous work for dimensional
crossover from 3D to 1D, the critical temperature is very
sensitive to the size and the shape of the system due to the
anisotropic growth of the correlation length. 9) In spite of the
long history of the Ising model, 11–13) Monte Carlo studies for
a dimensional crossover originating from the anisotropic
interaction are few. 9,10,14) As will be discussed below, a
marked finite size effect disturbs the Monte Carlo study as
goes to zero.
Monte Carlo simulations were employed for the classical
Ising spins, jsj ¼1, placed on an L L L simple cubic
lattice. Periodic boundary conditions were applied to
eliminate boundary effects. Wolff’s single cluster algorithm
was employed to update spin configurations. 15) All the
measurements were carried out by decreasing the temperature
from an infinite temperature. For thermal equilibration,
4000 MCSs (Monte Carlo Steps) were used. Measurements
were taken every 20 MCSs in order to avoid a correlation
between the measurements. As a result, 1000–100000
averages for a physical quantity were obtained.
The measured quantities were the magnetization M and
the susceptibility defined by
M ¼ 1
L3 * 2 ! 3
2
1=2+
X
4 SjðrÞ 5 ; ð3Þ
¼ 1
T
r; j
1
L3 2 * ! 2+
X
4 SjðrÞ
r; j
L 3 M 2
3
5; ð4Þ
where h...i indicates an ensemble average and the
Boltzmann constant is set to unity. The temperature of the
maximum susceptibility was adopted as an effective critical
temperature TcðL; Þ. We also measured the magnetization
and the susceptibility in a single chain defined by
Mr ¼ 1
L3 X
r
r ¼ 1
T
2
4
X
SjðrÞ
j
! 2
3
5
1
L3 2 * ! 2+
X X
4
SjðrÞ
r
j
1=2
; ð5Þ
L 3 M 2 r
LETTERS
3
5: ð6Þ
Mr and r indicate the magnetization and the susceptibility
in a chain. The temperature of the maximum r was adopted

2592 J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 LETTERS K. W. LEE
as an effective critical temperature TrðL; Þ in a chain.
Tcð Þ and Trð Þ are the critical temperatures in the
statistical limit, L !1. Trð Þ does not distinguish whether
a 3D long-range order or a 1D long-range order induced the
long range order in a chain. In the 3D limit ! 1, Trð Þ
becomes equal to the 3D Ising critical temperature, where a
3D long range order induces the long range order in each
chain. In the opposite limit ! 0, Trð Þ will be equal to the
1D Ising critical temperature, where a 1D long range order
induces the long range order in each chain, even in the
absence of a 3D long range order.
In a finite lattice, the effective critical temperature
corresponds to the temperature at which the correlation
length becomes equal to the lattice size. In a quasi-1D
system, the correlation length will grow anisotropically due
to the anisotropic interaction. Due to the weak interchain
interaction, the growth in the xy-plane is much slower than
the growth along each chain. TcðL; Þ, where a 3D long range
order occurs, is determined by the slowest growth in the xyplane,
whereas TrðL; Þ is determined by the growth along
the chain. Of course, in the statistical limit, those effective
critical temperatures should converge to a critical temperature,
because they reflect the same 3D long range order.
Therefore, Tcð Þ¼Trð Þ and TrðL; Þ TcðL; Þ always for
>0.
Figure 1 shows and r measured with L ¼ 10, where the
temperatures of maximum and r show the anisotropic
ordering and show that Trð10; Þ > Tcð10; Þ, as discussed
above. In the high-temperature region above a crossover
temperature, the quasi-1D system acts as a 1D system
because of the weak interchain correlation, 16) where the
susceptibility r in a chain should be the same as that in the
1D Ising model and thus should be independent of . Figure
1 shows that, in the high-temperature region, r is really
independent of , which remarkably shows the 1D behavior
of r other than .
Fig. 1. , and r as a function of J=T measured with L ¼ 10 and
¼ 0:05, 0.03, and 0.02 from the left. The dotted and solid lines
correspond to and r, respectively.
If there is a dimensional crossover temperature, 16) there
should be a crossover size Lcð Þ, where a dimensional
crossover occurs. The difference between TcðL; Þ and
TrðL; Þ decreases with L !1and increases with L ! 1.
In a small lattice, TrðL; Þ is far above TcðL; Þ and the
interchain spin-correlation can be neglected near TrðL; Þ.
Then TrðL; Þ will behave as an effective critical temperature
of the 1D Ising model. Near TcðL; Þ, each chain has already
been ordered with the magnetization M r ¼þ1 or 1 and the
quasi-1D Ising model behaves as a 2D system. When the
statistical limit is approached, the interchain spin-correlation
near TrðL; Þ cannot be neglected and the quasi-1D Ising
model recovers the 3D nature.
Figure 2 shows the measured TrðL; 0Þ as a function of L.
TrðL; 0Þ is expected to show the 1D Ising behavior because
of the absence of the interchain interaction. Although the 1D
Ising model should be ordered at zero temperature, a finite
size effect gives rise to a finite temperature magnetic
ordering in a finite lattice. In the 1D Ising model, the
correlation length can be expressed as 17)
e 2J=T : ð7Þ
The effective critical temperature TrðL; 0Þ corresponds to the
temperature where the correlation length becomes equal to
the lattice size, i.e., L. Then the effective critical
temperature TrðL; 0Þ has the following form:
TrðL; 0Þ
J
2
: ð8Þ
ln L
The dotted line in Fig. 2 corresponds to eq. (8) and the solid
line in Fig. 2 shows that Að2= ln LÞ with A ¼ 1:135. The
result shows that the finite size dependence of TrðL; 0Þ is well
described by eq. (8).
Figure 3 shows the measured TrðL; Þ as a function of .
When ! 0, TrðL; Þ leads to a constant value TrðLÞ, which
Fig. 2. Finite size dependence of TrðL; 0Þ. The dotted line corresponds to
eq. (8) and the solid line shows that Að2= ln LÞ with A ¼ 1:135. TrðL; 0Þ
follows the finite size dependence of eq. (8) well in the 1D Ising model.

J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 LETTERS K. W. LEE 2593
Fig. 3. Effective critical temperature TrðL; Þ vs . As decreases,
TrðL; Þ leads to a constant value TrðLÞ, which is independent of . The
constant TrðL; Þ¼TrðLÞ indicates that TrðL; Þ behaves as a 1D Ising
critical temperature.
is independent of . TrðLÞ has the same value as TrðL; 0Þ and
follows the finite size dependence of the 1D Ising model [eq.
(8)]. As discussed above, TrðL; Þ really undergoes a
dimensional crossover from 3D (quasi-1D) to 1D at a
certain crossover size Lcð Þ, a size below which the quasi-1D
Ising model behaves as a 2D Ising system.
Figure 4 shows Tcð20; Þ and Trð20; Þ as a function of .
Fig. 4. Tcð20; Þ and Trð20; Þ vs . The open and solid symbols
correspond to Trð20; Þ and Tcð20; Þ, respectively. The solid line
indicates the asymptotic form of eq. (2). A dimensional crossover of
Trð20; Þ appears as a departure from the asymptotic form of the quasi-1D
Ising critical temperature.
The solid line in the figure shows the asymptotic form of eq.
(2). For a large , which is still far from ¼ 1, the
asymptotic form of eq. (2) is consistent with Tcð20; Þ and
Trð20; Þ. As decreases, Tcð20; Þ and Trð20; Þ largely
deviate from the asymptotic form. When Trð20; Þ deviates
from the asymptotic form, it leads to a constant value TrðLÞ,
which is independent of . As shown in Fig. 3, a
dimensional crossover from 3D to 1D in Trð20; Þ appears
as the deviation. Then, by equating the 1D Ising effective
critical temperature of eq. (8) to the asymptotic form of eq.
(2),
2
ln Lc
2 ln 1
ln ln 1
we can estimate a crossover size Lc, where a dimensional
crossover occurs:
1
Lc : ð10Þ
ln
Below Lc, TrðL; Þ behaves as the effective critical
temperature of the 1D Ising model and the quasi-1D Ising
system acts as the 2D Ising model, as discussed above.
Below Lc, the quasi-1D Ising model behaves as the 2D
Ising model with a reduced 2D Ising Hamiltonian 18)
H ¼ JL X
M r M r0
:
ð11Þ
Then, the correlation length can be described as
r;r 0
T
L Tcð2DÞ
1
1
;
ð9Þ
1
; ð12Þ
where Tcð2DÞ is a 2D Ising critical temperature, and the 2D
Ising value ¼ 1 of the critical exponent for the correlation
length was used. Because the correlation length is equal to
the lattice size L at the effective critical temperature TcðL; Þ,
the critical tempreature can be expressed as
TcðL; Þ Tcð2DÞð1 þ LÞ : ð13Þ
Figure 5 shows the measured TcðL; Þ as a function of .
When ! 0, TcðL; Þ shows a linear dependence on ,
which is expected in eq. (13), and the solid lines in Fig. 5
show the linear fits. The inset shows the slope of the linear
fit. The solid line in the inset shows a fit to eq. (13) with
Tcð2DÞ ¼2:24, which is very similar to the exact 2D Ising
critical temperature Tcð2DÞ ¼2= lnð1 þ ffiffi p
2Þ¼2:269.
The
quasi-1D Ising model really undergoes a dimensional
crossover from 3D to 2D at the crossover size Lc.
In summary, a Monte Carlo simulation and a finite-size
scaling were employed to study the dimensional crossover in
the quasi-1D Ising model. The effective critical temperature,
defined as the temperature of maximum susceptibility,
showed a marked finite-size effect. The finite size effect
was elucidated to be due to a dimensional crossover from 3D
(quasi-1D) to 2D at the crossover size Lc 1=ð ln Þ.
Acknowledgments
This work was supported by Korea Research Foundation
Grant. (KRF-2001-002-D00111)

2594 J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 LETTERS K. W. LEE
Fig. 5. TcðL; Þ vs . The solid line shows a linear fit, whose slope is
shown in the inset as a function of L. The solid line in the inset shows a fit
to eq. (13) with Tcð2DÞ ¼2:24. The linear dependence on indicates that
TcðL; Þ behaves as a 2D Ising critical temperature with a reduced 2D
Ising Hamiltonian of eq. (11).
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