3D Ising Nonuniversality: a Monte Carlo study

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by  Melanie Schulte

3D Ising Nonuniversality: a Monte Carlo study
5
0
0
2
Melanie S
hulte and Caroline Drope
n
a
J Institute for Theoreti
al Physi
s, Cologne University, 50923 Köln, Germany.

9 Email: msthp.uni-koeln.de, dropethp.uni-koeln.de.
2

]
h
Abstra
t
c
e
m Weinvestigate asamember oftheIsinguniversality
lasstheNext-Nearest
Neighbour Ising model without external (cid:28)eld on a simple
ubi
latti
e by
-
t
a using the Monte Carlo Metropolis Algorithm. The Binder
umulant and the
t sus
eptibility ratio, whi
h should be universal quantities at the
riti
al point,
s
t. were shown to vary for small negative next-nearest neighbour intera
tions.
a
m
keywords: next-nearest neighbour Ising model, Binder
umulant, fourth order
u-
-
d mulant.
n
o
c
[ 1 Introdu
tion

1
v The greatvariety of
riti
albehaviour isredu
ed by dividingallsystems intoa small
1
number of universality
lasses. These
lasses are
hara
terized by global properties
3
7 like dimensionality and the number of
omponents of the order parameter and have
1
the same set of
riti
al exponents and s
aling fun
tions, whi
h are independent of
0
5 mi
ros
opi
details[1℄. Here the3D Ising universality
lassis
onsidered by studying
0
the Binder
umulant [2℄ of the Next-Nearest Neighbour Ising model (NNN model).
/
t
a If alluniversal quantities of one
lass are known, the asymptoti

riti
albehaviour is
m
believedtobealsoknown
ompletelyunderthe
ondition,thatonlytwononuniversal
K K
- amplitudes t and h are given. This parti
ular feature is
alled two-s
ale fa
tor
d f
n universality. The free-energy-density , from whi
h we
an derive thermodynami
al
k
o B
quantities and whi
h is measured in units of ,
an be expressed by [3℄:
c
: f(t,H;L) = f (t,H;L)+f (t;L),
v s ns
i
X
t = 1 T/T H f
c s
r where − and is the ordering (cid:28)eld. The singular part is the part
a L fns
whi
h yields thermodynami
al singularities in the → ∞ limit. denotes the
f
non-singular part of , whi
h
an be
hosen without (cid:28)eld dependen
e. For the
f
s
singular part we
an write for the Ising model
f (t,H;L) = L−dY(K tL1/ν,K HLδ/ν)
s t h
Y(x,y)
is
alled the s
aling fun
tionand isuniversal using system-dependent fa
tors
K K
h t
and .
Chφe4n and Dohm [4℄ re
ently predi
ted some deviations from this universality, from
a theory.
In the following se
tion we study the universal value of the Binder
umulant for
1
J
NNN
the NNN model for di(cid:27)erent next-nearest neighbour intera
tion by simula-
tions. A

ording to Privman, Hohenberg and Aharony [3℄, the Binder
umulant
s
ales a

ording to
U (t,L) G(K tL1/ν) ,
4 t
≈
G(x) = [(∂4Y/∂y4)/(∂2Y/∂y2)2] t = 0 T = T U (0,L)
y=0 c 4
where . For ∗ (mean∗s ),
L U (0, ) U (0,L)
4 4
approa
hes with growing a universal
onstant ∞ , whi
h∗we are
U (0, )
4
going to simulate, shows only small deviations from the universal value ∞
[5∗℄. Consequently, it is possible to give a realisti
approximation of the (cid:28)nal value
U (0, )
4
∞ simulating (cid:28)nite latti
es.
2 Applied NNN Models
2.1 Standard Next-Nearest Neighbour Model
∗
U
4
To show the variation of within the 3D Ising universality
lass, we
hoose the
Next-Nearest Neighbour model (NNN Model) without external (cid:28)eld and a simple
a

ubi
latti
e. Ea
h latti
e site has 6 next neighbours in the distan
e and 12 next-
√2 a a
nearest neighbours in the distan
e · , if is de(cid:28)ned as the latti
e
onstant. The
J
NNN
ex
hange for
e of the next-nearest neighbours
an now be
hosen to be di(cid:27)er-
J
NN
ent from the ex
hange for
e of the 6 nearest neighbours as we also would also
intuitively expe
t, be
ause their distan
e to the
onsidered site is also not identi
al.
A ferromagneti
phase transition is also existent for an antiferromagneti
ex
hange
J
NNN
for
e . This kind of ex
hange, ferromagneti
for nearest neighbours and anti-
ferro∗magneti
for next-nearest neighbours, will be treated for our universality test
U
4
of .
2.2 Anisotropi
Next-Nearest Neighbour Model
∗
U
4
Furthermore we studied the universality of in the zero (cid:28)eld NNN Ising Model
J > 0
with an isotropi
nearest-neighbour (NN)
oupling and an anisotropi
next-
J < 0
NNN
nearest neighbour (ANNN)
oupling . We refer to the
onsidered model
of Chen and Dohm in their publi
ation (cid:16)Nonuniversal (cid:28)nite-size s
aling in anistropi
systems(cid:16) [4℄. The anisotropi
NNN Ising Model is established by
onsidering only 6
J < 0
NNN
of the 12 next-nearest neighbours being e(cid:27)e
tive for NNN intera
tion and
the other 6 NNN having no NNN intera
tion. The e(cid:27)e
tive next-nearest neighbours
(1,0,0) (1,0,1) (0,1,1)
have the positions ± , ± and ± of a simple
ubi
latti
e.
3 Simulation Results
J J = 1
NNN NN
Simulatingthe NNN model
anbe varied, whileitis
hosen . Forval-
J J 0.5
NNN NNN,crit
ues of higher than a
riti
al value ≈ − we observe ferromagneti
J
NNN,crit
phase transitions, while below no ferromagneti
phase transition o

urs.
J = 0
NNN
At we obtain the standard 3-dimensional (3D) Ising ferromagnet. Here
J < J < 0
NNN,crit NNN
the region and thus a small antiferromagneti
intera
tion
J
NNN
is
onsidered.
2
Si∗mulating now the "(cid:28)xed po∗int value" at the
riti
al point of the Binder
umulant
U = U (T ) U (J )
4 4 c 4 NNN
, we evaluate to
he
k its universality within the 3D Ising
universality
lass. For an Ising Model in zero (cid:28)eld, the Binder
umulant simpli(cid:28)es
to
M4
U = 1 h i ,
4 − 3 M2 2 (1)
h i
∗
M U
4
where symbols the magnetization per spin. For the determination of ea
h
U T L
4
was plotted as a fun
tion of temperature for three di(cid:27)erent∗latti
e lengths
= 10 20 40 U (T)
4
( , , ). To get more pre
ise results, several
urves of for di(cid:27)erent
latti
e sizes with di(cid:27)erent seeds were averaged. In (cid:28)gure 1 is shown this method for
J
NNN
two di(cid:27)erent . The errors were
al
ulated by averaging the values of the three
L = (10,20) (10,40) (20,40)
di(cid:27)erent
rossing points of , , .
0.6
J =0.000 0.6 J =-0.247
NNN NNN
0.55
0.55
0.5
0.5
U4 0.45 U4
0.45
0.4
0.4
0.35 0.35
0.3 0.3
4.5 4.504 4.508 4.512 4.516 4.52 1.22 1.222 1.224 1.226 1.228 1.23
T [J /k ] T [J /k ]
NN B NN B
∗
U T J
4 c NNN
Figure 1: Lo
alization of the (cid:28)xed point ( , ) for di(cid:27)erent by quantifying
U (T) L = 10 20 40
4
the
rossing point of for latti
e sizes (dotted), (dashed) and
(solid).
3.1 Results for the Standard NNN Ising Model
∗
U (J )
4 NNN
In (cid:28)gure 2 and table 1 we
an see all simulated results as a fun
tion and
these results in
om∗parison with theoreti
al results of Chen and Dohm [4℄ plotted
1 U (w)/U(0) w = J /(J + 2J )
4 NNN NN NNN
as a fun
tion − with . Chen and Dohm

onsidered an anisotropi
NNN ∗Ising model, whi
h we will dis
uss later. It
learly
U J
4 NNN

an be seen in(cid:28)gure and, that isnota (cid:28)xed value fordi(cid:27)erent , parti
ularly
J =
NNN
in the interval of [-0.25:-0.22℄. This implies in this region the absen
e of
universality for the 3D Standard NN∗ N Ising model.
U L J = 0
4 NNN
Great deviations of the resulting for → ∞
an be ex
luded for [5℄.
L = 100 J = 0
NNN
Observing latti
e length up to for 6 we observed no signi(cid:28)
ant

hange of our simulation results.
J
NNN
We
an
on
lude, that in the region of small negative values for the 3D Next-
Nearest Neighbour Ising model shows a di(cid:27)erent universal behaviour than the 3D
Nearest Neighbour Ising model.
3
∗ ∗ ∗
J w U 1 U (w)/U (0)
NNN 4 4 4
−
0.000 0.000 0.4853 3.20 10−3 0.0000 1.32 10−2
0.100 0.125 0.4867±4.80·10−3 2.8609 10−3 ±1.98·10−2
− − ± · − · ± ·
0.200 0.333 0.4894 6.90 10−3 8.4366 10−3 2.84 10−2
− − ± · − · ± ·
0.230 0.426 0.4854 1.68 10−3 2.6201 10−4 6.92 10−3
−0.240 −0.462 0.4753±3.22·10−3 −2.0685· 10−2± 1.33· 10−2
− − ± · · ± ·
0.241 0.465 0.4801 6.44 10−3 1.0700 10−2 2.65 10−2
− − ± · · ± ·
0.245 0.480 0.4692 1.02 10−2 3.3247 10−2 4.20 10−2
−0.246 −0.484 0.4290±1.40·10−2 0.1161· ±5.77·10−2
−0.247 −0.488 0.3994±3.96·10−3 0.1771 ±1.63·10−2
− − ± · ± ·
∗
U
4
Table 1: Simulation results of the universal quantity and its statisti
al errors for
J
NNN
di(cid:27)erent next-nearest neighbour intera
tion .
0.2
0.5
0.15
0.48
0)
0.46 (4 0.1
U*
U*4 0.44 w)/
(4
U* 0.05
0.42 1-
0
0.4
0.38 -0.05
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 -0.5 -0.4 -0.3 -0.2 -0.1 0
JNNN w
Figure 2∗: Standard 3D NNN model. Left side: Simulation results of the (cid:28)xed
U4 JNNN (cid:4)
point as a fun
tion of ; Right side: Results of the (cid:28)gure on the left ( )
in
omparison with theoreti
al results (for a di(cid:27)erent neighbourhood) of Chen and
Dohm [4℄ (dashed line).
3.2 Results for the ANNN Ising Model
Figure 3 and table 2 show the results for the ANNN Ising Model analogue to se
tion
3.1. The simulation results show a signi(cid:28)
ant non-universality of the Binder
umu-
lant. As
ompared to the analyti
results of Chen and Dohm [4℄ the analyti
al
urve
have a di(cid:27)erent te∗nden
y and is not within the errorbars of the simulation results.
w 0.5 U
4
For → − , in
reases whereas Chen and Dohm obtained a de
rease. Thus,
the theoreti
al results
annot be
on(cid:28)rmed fully.
4
0.52 0.12
0.515 0.1
0.51 0.08
0.06
0.505
0.5 (0)4 0.04
U 0.02
*U4 0.495 (w)/4 0
0.49 U
1- -0.02
0.485
-0.04
0.48 -0.06
0.475 -0.08
0.47 -0.1
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1
w w
+ U∗(w)
4
Figure 3: Left side: Simulation∗ results∗ ( ) for the ANNN model of . Right
1 U (w)/U (0)
4 4
side: Simulationresultsof − in
omparisonwiththetheoreti
alresults
(for this neighbourhood) of Chen and Dohm [4℄ (solid line).
∗ ∗ ∗
J w U 1 U (w)/U (0)
NNN 4 4 4
−
0 0 0.478 3.53 10−3 0 1.04 10−2
± · ± ·
0.143 0.2 0.484 5.80 10−3 1.33 10−2 1.43 10−2
− − ± · − · ± ·
0.188 0.3 0.488 6.04 10−3 2.06 10−2 1.47 10−2
−0.222 −0.4 0.498±5.96·10−3 −4.20·10−2 ±1.47·10−2
−0.237 −0.45 0.499±9.42·10−3 −4.50·10−2 ±2.12·10−2
− − ± · − · ± ·
0.245 0.48 0.507 7.08 10−3 6.07 10−2 1.68 10−2
− − ± · − · ± ·
0.247 0.49 0.507 1.10 10−2 5.82 10−2 2.43 10−2
−0.249 −0.495 0.507±8.34·10−3 −5.98·10−2 ±1.91·10−2
− − ± · − · ± ·
∗
U (w)
4
Table 2: Simulation results for the ANNN Model for .
5
3.3 Universality Test of the Sus
eptibility for the ANNN
Model
χ = (N/k T)(cid:0) M2 M 2(cid:1)
B
Simulating the sus
eptibility h i−h| |i for the ANNN Ising
J
NNN
Model we use the same range of NNN intera
tion . We
onsider two temper-
Tc T− = Tc 3%Tc T+ = Tc +3%Tc
atures near the
riti
al temperature , − and . The
results show expli
itly the absen
e of universality. Figure 4 shows the ratio of the
T+ T− w
sus
eptibilityat and asafun
tionof . Asigni(cid:28)
antdeviationofnearly100%
w = 0 w = 0.45 T L
c
exists here between and − . Varying below the latti
e size be-
10 317 MCS 50 5 105
tween and and the number of MoLnt=e C40arlo steMpsCS = 5be1tw05een and ·
we
an ex
lude (cid:28)nite-size e(cid:27)e
ts with and · . Table 3 shows
χ(T+) χ(T−) χ(T+)/χ(T−) JNNN
all results of , and for di(cid:27)eren<tM2 >. (W<eMlook>e2d at the
(cid:29)u
tuations in the absolute value of the magnetization − | | , not of
T 1 2/π = 0.3634
c
the magnetization. Thus our values above are lower by a fa
tor −
< M2 >
than the usual , assuming Gaussian (cid:29)u
tuations [2℄.)
4.5
4
T)- 3.5
χ(
)/+
T
χ( 3
2.5
2
-0.5 -0.4 -0.3 -0.2 -0.1 0
w
χ(T+)/χ(T−)
Figure 4: Simulation results of the ratio of sus
eptibility as a fun
tion
J
NNN
of .
JNNN w χ(T+) χ(T−) χ(T+)/χ(T−)
0 0 32.10 0.39 15.03 9.3 10−2 2.14 2.9 10−2
0.143 0.2 34.26±0.70 15.37±4.1·10−2 2.23±4.6·10−2
− − ± ± · ± ·
0.188 0.3 35.46 0.59 15.34 11.1 10−2 2.31 4.2 10−2
− − ± ± · ± ·
0.222 0.4 38.54 0.47 14.91 12.2 10−2 2.59 3.8 10−2
−0.237 −0.45 39.53±0.55 14.72± 6.6 ·10−2 2.69±3.9·10−2
− − ± ± · ± ·
0.245 0.48 57.27 1.36 14.28 12.5 10−2 4.01 10.2 10−2
− − ± ± · ± ·
T− = Tc 3%Tc T+ =
Table 3: Simulation results of the sus
eptibility at − and
Tc +3%Tc χ(T+)/χ(T−) JNNN
and of the ratio as a fun
tion of .
6
4 Con
lusions
This work examined the universality of the Binder
umulant for the 3D NNN Ising
model in the region of small negative values for the next-nearest neighbour intera
-
tion. Monte Carlo simulations of the universal quantity of the Binder
umulant and
the universal sus
eptibility ratio with di(cid:27)erent next-nearest neighbour intera
tions
J
NNN
showed a variation of both quantities for all applied models. The variation of
these universal quantities leads to the
on
lusion, that in the
onsidered region the
NNN Ising model does not belong fully to the 3D Ising universality
lass.
J J
NN NNN
We assume that for regions, where
hosen values of and still
ause fer-
romagneti
phase transitions, but are
lose to a region where we will (cid:28)nd no more
ferromagneti
behaviour, a new de(cid:28)nition for universality
lasses for this model has
to be found.
5 A
knowledgements
We thank Dietri
h Stau(cid:27)er for
ontributing his knowledge and experien
es in Monte
Carlo simulations. Spe
ial thanks also to Volker Dohm from the te
hni
al university
of Aa
hen and Xiaosong Chen from the Institute of Theoreti
al Physi
s, Chinese
A
ademy of S
ien
es for their dis
ussions and help.
Referen
es
[1℄ D. P. Landau and K. Binder, A Guideto Monte Carlo SimulationsinStatisti
al
Physi
s (Cambridge University Press, 2000).
[2℄ K. Binder, Z. Phys. B 43, 119 (1981).
[3℄ V. Privman, P. C. Hohenberg and A. Aharony, Phase Transitions and Criti
al
Phenomena 14, eds. C. Domb and J. L. Lebowitz (A
ademi
Press Limited
1991).
[4℄ X. S. Chen and V. Dohm. Phys. Rev. E 70, 056136 (2004).
[5℄ A. M. Ferrenberg, D. P. Landau, Phys. Rev. B 44, 5081 (1991).
7