- Email: [email protected]
Critical temperature of the non-frustrated ferromagnetic ising model on the quasiperiodic octagonal tiling
]OURN,,I L OF
NON¢ gT,T,T S0IN ELSEVIER
Journal of Non-CrystallineSolids 191 (1995) 216-226
Critical temperature of the non-frustrated ferromagnetic Ising model on the quasiperiodic octagonal tiling D. Ledue *, J. Teillet LMA URA CNRS808, Facult~des Sciences, Universit~de Rouen, 76821 Mont-Saint-Aignanc~dex, France
Received29 June 1994;revisedmanuscriptreceived22 March 1995
Abstract The critical temperature of some non-frustrated ferromagnetic Ising spin systems on the two-dimensionaloctagonal tiling is calculated by Monte Carlo simulations using the simulated annealing method. The ferromagnetic interactions are limited to the third neighbours (J1, ./2, J3) where only J2 is a percolating interaction. The infinite-tiling critical temperature is estimated at kTc/J = 2.39 + 0.02 (J2 = J and Jl = J3 = 0), kTe/J = 3.05 --I-0.03 (J2 = Jl ~=J and J3 = 0), kTe/J = 3.60 + 0.04 (J2 = J3 = J and J1 = 0) and kTJJ = 4.39 + 0.04 (./2 = J1 = J3 = J). The value of kTo/(z) J is generally slightly higher in the octagonal tiling than in the square and triangular lattices indicating that the tendency to ferromagnetic ordering is higher in quasiperiodic tilings. It is found that the critical temperature, Tc, varies linearly with the different exchange integrals.
1. Introduction Some experiments on the magnetism of the icosahedral alloys have shown spin-glass-like properties at very low temperature for AI-Mn and AI-Mn-Si (see, for example, Refs. [1-3]). Later, weak ferromagnetic quasicrystals were claimed to be found [4,5]. In particular, A14oMn25F%CuvG%5 is ferromagnetic at 300 K and the Fe atoms have a weak moment (0.11/z B) [4]. Other investigations have shown ferromagnetic quasicrystals with higher Curie temperature (Tc = 533 K for A1525Ge22.sMn25) [5]. On the other hand, while classical spin systems on two-dimensional (2D) periodic lattices have been
* Correspondingauthor. Tel: +33 35 14 68 77. Telefax: +33 35 14 66 52. E-mail: denis.ledue@univ_rouen.fr.
widely studied (see, for example, Refs. [6-8] for the Ising model and Refs. [9-11] for the XY model), only a few studies have been devoted to 2D quasiperiodic systems [12-14]. For the XY model, numerical simulations on the Penrose tiling provided some frustrated magnetic ground states and indicated a Kosterlitz-Thouless-like transition [12]. For the frustrated Ising model on the octagonal tiling, analytical calculations have indicated some long-rangeordered magnetic ground states [13]. Also in the Penrose tiling, the magnetic transition of nonfrustrated ferromagnetic Ising spin systems has been studied using numerical simulations [14-16]. The results show that the Ising model on the Penrose tiling belongs to the same universality class as the Ising model on periodic lattices. Further, the critical temperature of the infinite Penrose tiling has been estimated as k T c / J = 2.401 + 0,005 [14] which is
0022-3093/95/$09.50 © 1995 ElsevierScienceB.V. All rights reserved SSDI 0022-3093(95)00246-4
D. Ledue, Jr. Teillet/ Journal of Non-Crystalline Solids 191 (1995) 216-226
higher than the critical temperature of the infinite square lattice ( k T c / J = 2.269 [6]) with the same mean number of interacting neighbours (( z ) = 4). In this paper, we present calculations of the critical temperature of the Ising model on another infinite quasiperiodic tiling: the octagonal tiling. Our study is carried out by numerical simulations and the data are compared with analytical results previously obtained on periodic lattices [17,18]. The octagonal tiling exhibits six different types of local environment (Fig. l(a)). The ferromagnetic ex-
217
t
a) Fig. 2. Some clusters due to JI or J3 (0, spins which belong to a Jl-cluster of four spins; [3, spins which belong to a Jl-cluster of eight spins; 0, spins which belongto a J3-clusterof 24 spins; O, spins which belong to a J3-clusterof 72 spins).
A
B
C
D
E
F
Fig. 1. (a) The octagonal tiling and the six local environments. (b) The exchange interactions (J2 correspondsto the edges of the unit cells).
change interactions have been limited to the third neighbours (J1/> 0, ,/2 >~ 0, J3 >~ 0) (Fig. l(b)). Unlike periodic lattices, all the exchange interactions do not allow magnetic percolation: only J2 favours alignment of all spins while J1 and J3 can only connect the spins into ferromagnetic clusters (Fig. 2). Then, we can deduce that the ground states are (i) ,/2 ~ 0: ferromagnetic ground state; (ii) ,/2 = 0: (a) J1 va 0 and "/3 4~ 0: ground states made up of two disconnected ferromagnetic sublattices and isolated spins (sites A) (Fig. 3); (b) J1 -- 0 and "/3 va 0: ground states made up of ferromagnetic clusters and isolated spins (sites A); (c) J1 4~ 0 and J3 = 0: ground states made up of small ferromagnetic clusters (maximal size: eight spins) and isolated spins (sites A, B, C, D). Different sets of positive exchange integrals on finite octagonal tilings have been used in order to study the influence of each exchange integral. The models, simulation technique and data analysis method are described in Section 2. The results and a discussion are presented in Section 3.
218
D. Ledue, J. TeiIlet / Journal of Non-Crystalline Solids 191 (1995) 216-226
2. Background 2.1. The octagonal tiling The quasiperiodic octagonal tiling exhibits an eight-fold non-crystallographic symmetry [19]. The 4D integral representation of C s, the eightfold cyclic group, decomposes into two inequivalent 2D representations: one operates on EEl, the physical space, the other on the orthogonal complementary space,
E.L. The 2D octagonal tiling is obtained by projecting the nodes of the original lattice, 774, which have a projection in E . falling within the octagonal window [13]. All the octagonal tilings belong to the same local isomorphism class, therefore the infinitetiling critical temperature does not depend on the tiling [20]. So, we have considered the octagonal tiling with a perfect eightfold symmetry around its centre. This tiling is generated using in E± an octagonal window which is centered on the origin. 2.2. Model and magnetic quantities
a)
Let each vertex of the octagonal tiling be occupied by an Ising spin, S t = + 1. The magnetic energy, U, and total magnetization, M, (in g/z B, where g is the Land6 factor and /xB is the Bohr magneton), are respectively given by
v=-
E s,is,sj, (i,j)
M = ~ S i. i The thermal capacity, C(T), is deduced from the fluctuations in the magnetic energy: c(r)
b)
(u 2 ) - ( U ) 2
ou
kT 2
OT'
=
where 1
(U) =
Z(T)
EV(c) exp(-V(c)/kT) ¢
is the thermal average (Z(T) = Ec e x p ( - U(c)/kT) is the partition function and the sum goes over all configurations, c). 2.3. Numerical simulation
Fig. 3. Ground states with Jl 4= 0, J3 @ 0 and '/2 = 0: (a) the two sublattices in opposite directions, (b) the two sublattices in the same direction (QIDO, sites A; e, sites B; ©, sites C; n, sites
D; 1,, sites E).
The procedure is a simulated annealing method [21]. In this Monte Carlo (MC) procedure, the magnetic energy is minimized by adjusting the direction of the spins in such a way as to freeze the system into a magnetic ground state. At each temperature, thermal equilibrium is achieved after a few thousands MC steps per spin following the Metropolis algorithm [22]. The spins are examined one after another and the energy variation, AU, which corre-
D. Ledue, J. Teillet / Journal of Non-CrystaIIine Solids 191 (1995) 216-226
219
(b) o.7
(8_) 0.7
i
0.6-
0.6 0.5 0.5
•
0,4 v
O o
o
0,4
• 0.3
o
••
t o
o
o
o
•4361
o
O o
o ,:3
0.2
O
e••
01
• 1433
0
O o
002481
• ••• t ••••
0,1
÷
f o
0.2
• 0.3
•4361
too
o
o
02929
'i'
::iii
O{
•
O O O O
•1169
o o o 0689 0
oo481
J
0 2
2,5
2.5
3
3.5
3
kT/J
kT/J
(d)
(O) 0.6
i.
0.5
0.6
0,5
o 0.4
0,4
°i
° o
0
o
o
•
•
•
o o
0.2
o
O
0,2
o
o
(3
O
o
o
-
O
o2929
o
o o
02929
0.1
• °1L• •
0,1
:: 0
0 2.5
3
3.5 kT/J
•4361
O
•4361
O
O
o
•
0
0 •
o
o
>~ 0,3
0.3
l°
4
3.5
o 0 o o
f
o o o
•
•1169
o o o 0689
;
J
I
4
4.5
5
kT/J
Fig. 4. Temperature variation of the thermal capacity for several lattice sizes ((a) (0, J, 0), (b) (J, J, 0), (c) (0, J, J), (d) (J, J, J)).
220
D. Ledue, J. TeiIIet/ Journal of Non-Crystalline Solids 191 (1995) 216-226
4.5
~
4 ° ~ 0
0
3.5 ?~"--~ " 3
~'(:~:X>
*""*'~
~
,
,
~o(J,J,J)
+ ' ' " ~
~ * ( O , J , J ) o(J,J,O)
+~°~°"
O"----.
2,5 2 1.5
-&(O,J,O) I
~
I
4
0.05
0,1
0.15
0,2
1,1"1/2 Fig. 5. Extrapolationof kTc(n)/J v e r s u s n -1/2.
sponds to the spin-flip is calculated. If the spin-flip decreases the magnetic energy of the system (AU ~< 0), it is accepted. If it increases the energy (AU > 0), it is not systematically rejected: the transition occurs with probability W ( A U ) = e x p ( - A g / k T ) . The low-temperature equilibrium is reached by simulated annealing: during the process, T is slowly decreased and, if this annealing is performed slowly enough, the spin configuration which is frozen will correspond to one of the ground states. Non-frustrated spin systems are ergodic, i.e., at thermal equilibrium, the average over MC steps converges towards the thermal average. For each run, 5000 MC steps per spin have been discarded before averaging U and M over the next 15 000 MC, steps per spin. Numerical simulations were carried out on finite octagonal tilings of different sizes (n = 334361) with free boundary conditions as in Ref. [12].
Table 1 Infinite-tilingcritical temperaturein the four cases of exchange interactions (mean numbers of interactingneighboursare analytical results) Jl .I2 J3 (z) kTc/ J 0 J 0 4 2.39,--0.02 J S 0 (4+ 6~-)/(1 + ~-) 3.05,--0.03 0 J J (8+ 4~')/(i+ ~ ) 3.60,-,0.04 J J J (8 + 6~')/(1 + V~") 4.39 _ 0.04
The cooling rate, ~-, was choosen equal to 0.98 so that simulations provide regular thermal capacity curves for each size. Numerical simulations were performed on a DG MV 15000-20 computer. 2.4. Calculation method of the critical temperature To determine the infinite-tiling critical temperature, we have used the 'finite-tiling transition temperature', To(n), which corresponds to the maximum in the thermal capacity or, equivalently, the point of maximum slope in magnetic energy [23-27]. The infinite-tiling critical temperature can be calculated by extrapolating the asymptotical linear variation of Table 2 Comparisonof kTc / ( z) J in quasiperiodictilingswith kTc / ( z ) J in periodiclattices Lattice (z) kTe / (z) J Square 4 0£67 [6,17,18] 8 0.657 [17] Triangular 6 0,608 [17] 12 0.731 [17] Penrose 4 0.600::1:0.001 [14] Octagonal 4 0.5974-0.006 (4+ 6V~-)/(I+ ~ ) 0,5904-0.006 (8+4V~')/(I+ ~ ) 0.636:I:0.006 (8+ 6~')/(I + f2") 0.643+-0,006
D. Ledue, J. TeilIet/ JournaI of Non-Crystalline Solids 191 (1995) 216-226
QP lattices
o
P lattices
0.9
•
Beth• theory N
221
- ' " ""
,,.,''* oo
.o•
0.8
....--
..-'''""
v
0.7
..'" o ._o2, ~
0.6
A/ l
I
I
l
I
I
2
4
6
8
10
12
0.5 0
Fig, 6, Comparisonof kTc/(z) J in quasiperiodic filings with kTc/{ z)J in periodic lattices. O- J2= J / 4 , 0 - J 2 = 3 J / 4 , O - J 2 = J , o - J 2 = 3 J / 2
(a) 0.7 0.6
O0
@O
0.5 > o
O O
',
O
**• • 0.
•
0
O
•
O
O
0
0.4 #
0.3
0 O O0
•
0• ,00 •
0
• 000
~o~O0 ° •*
00 0 0
%
0.2 0.1 0
,
1
I
E
I
I
l
I
2
3
4
5
6
7
kT/J • - J2 = J / 4 , <>- J2 = 3 J / 4 , O- J2 = J, O- J 2 = 3 J / 2
(b) 1
0
•
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 o.1
*
0
OOo ".... °°Ooo
\ •
0
0 •
0
•
0
0
0 <;,
•
0
}..
0
o 1
1
I
2
3
• •
<>
•
°
•
00
0 •
4
5
o° ooo 6
•
o
o
o
i
i
7
8
kT/J
Fig. 7. Temperature dependence of (a) the thermal capacity and (b) the order parameter for several values of J2 (finite octagonal tiling of 4361 spins).
222
D. Ledue, J. TeilIet/ Journal of Non-CrystallineSolids 191 (199512•6-226
To(n) versus n -llz~ (v is the critical exponent associated to the correlation length) [6]. Numerical simulations on the ferromagnetic Ising model on the octagonal tiling provided v = 1 which is the 2D periodic value [28]. For a given size and a given set of exchange integrals, it has been shown that several numerical simulations provide similar thermal capacity curves. So, the accuracy on the infinite-tiling critical temperature is not limited by the statistical fluctuations from one numerical simulation to another but by the accuracy in the estimate of the location r~(n) of the maximum in the thermal capacity. Then, the uncertainty has been estimated at ATo = ½(1 - 'r)T~, where ¢ is the cooling rate.
3. Results
tiling is generally higher than in the square and triangular ones (Fig. 6). This difference can be explained assuming that quasiperiodicity is less fair to the propagation of the spatial fluctuations of the local magnetic field than periodicity. It should be noted that, according to the accuracy of our data, we cannot conclude if adding the nearest neighbour interaction, J1, changes the value of k T d ( z ) J (Fig. 6 and Table 2). This look of change can be explained by the fact that J1 has a small influence on the transition temperature because it can only connect the spins into small ferromagnetic clusters (maximal size: eight spins). On the other hand, taking into account the interaction with the third neighbours, J3, increases k T c / ( z ) J although J3 (as Jl) can only connect the spins into ferromagnetic clusters (nonpercolating interactions). This effect is attributed to the mean size of the clusters due to J3 (J3 can
As expected, since the transition should be ferromagnetic (J2 @ 0), C / k ~ 0 as T ~ OK ( C / n k 1/2 for the Kosterlitz-Thouless transition [12,29]). The size dependence of the order parameter data ((IMI)n) is quite pronounced even well below T~(n) as in the square lattice with free edges [6].
• - vs J2/J ~
/ .,,.,,X
,Di~-"~ 7 x
The infinite octagonal tiling critical temperature has been calculated in four cases which correspond to four sets of exchange interactions (J1, J2, -/3): The temperature variation of the thermal capacity for several lattice sizes indicates that the maximum in the thermal capacity, Cm,~×, increases with the lattice size (Fig. 4). The asymptotical behaviour of To(n) versus n -I/2 ( n - I / 2 ~ 0) is shown in Fig. 5. The infinite-tiling critical temperature for each case is given in Table 1. If ( z ) = 4, the octagonal tiling and the Penrose tiling exhibit close critical temperatures [14]. However, since the octagonal tiling and the Penrose tiling are not locally isomorphic, we cannot conclude that these temperatures are equal. In order to study the influence of the quasiperiodicity, the values of k T c / ( z ) J for several periodic and quasiperiodic lattices are regrouped in Table 2. Our simulations show that k T o / ( z ) J in the octagonal
/
x . vs J3/J
3.1. Infinite-tiling critical temperature - - comparison between periodic and quasiperiodic lattices
(o, J, 0), (J, J, o), (o, s, J), (J, s, s).
/ *
-VSJl/J
4 . -'Z''"
×1 i
,/
f 3
1//~l'/
×
0
/
/
l,
I-
I
I
1
1
I
0.25
0,5
0,75
1
1,25
1.5
Fig. 8. Variationof kTe/J versus the exchangeintegrals (finite octagonaltiling of 4361 spins).
D. Ledue, J. TeilIet/ Journal of NoniCrystalline Solids 191 (1995)216-226
223
3.2. Influence of each exchange integrals (n = 4361)
connect the spins into large clusters which surround the centre of the tiling) (Fig. 2). As expected, our values of k T c / ( z ) J are lower than those given by the Bethe theory (kTc/zJ = 2 / ( z In(z/z - 2))) [30], which partially neglects the local magnetic field fluctuations near the transition (Fig. 6). It should be noted that in the square lattice with ( z ) = 8, a lower value k T J ( z ) J = 0.620 has been found by Gibberd [18]. Unlike Ref. [17] in which the authors have used series expansion, Gibberd has considered the freefermion approximation which has been used previously in the 2D ferroelectric problem.
In this section, calculations are restricted to a decent-sized finite octagonal tiling ( n - - 4 3 6 1 ~ [T~ - T~(4361)]/T~ - 0.02) (Fig. 5). The transition temperature has been determined in three cases in order to study the influence of each exchange integral: (i) influence of ,/2: J 1 = J3 = J and J / 4 <~J 2
3J/2 (the case '/2 = 0 which is slightly different has been excluded (see Section 1)); (ii) influence of Jl: 72 = J3 = J and 0 < JI ~< 3J/2;
X - J1 = O, • - J1 = J / 4 , O - J1 = 3 J / 4 , 0 - J1 = 3 J / 2
(a) 0.7 0.6
•
× X
0,5
X
O
O< • X
0.3
x
XXo
0 o
XO
•
0
0
0 0
0 0
X
<> 0 •
0
0
o,
•
X
0
0
•
,,
0,4
o
0
0 0
oOO
X
• X
¢
•
0
0
0.2 0.1 0
3
I
I
3.5
4
l
I
I
4,5
5
5.5
kT/J x- J l = 0 , O - J l = J / 4 , O -
Jl=3J/4, o- Jl=3J/2
(b) I X •
0.9
X@
~
0.8
OOt~0 0000,~^ •~OO 0 ~uO 0
Xx~',o % ^
0.7
"'Xx • X
0.6
X
Oo
~O •
UO O
0
•
0
0
X
i o.5
,5
0.4 0.3 0.2 0.1 0
I O
•
0
X @
0
Xx~l~,• • I P
2
3
4
,"
@<> X i
5
00¢><
0
6
O E
7
kT/J
Fig. 9. Temperature dependence of (a) the thermal capacity and (b) the order parameter for several values of Jl (finite octagonal tiling of 4361 spins).
D. Ledue,J. Teillet/ JournaIof Non-CrystallineSolids191 (1995)216-226
224
(iii) influence of J 3 : J 2
Influence of JI (J2 = "Is = J) Increasing J1 does not broaden the peak of the thermal capacity (A = 0.9) and does not change the tail of the order parameter above T~ (Fig. 9) because the interaction length of the NN interaction, Jl, is
3,2.2.
= J1 = J and 0 ~
3:/2. 3.2.1. Influence o f J 2 (J~ = J3 = J) The temperature dependence of the thermal capacity and the order parameter for several values of J2 are drawn in Fig. 7. Increasing J2 broadens the peak of the thermal capacity (0.5 ~< A ~ 1.3; A is the width at half-height) and is responsible for a more pronounced tail of the order parameter at high temperatures. This tail is attributed to the short-range magnetic order above T~. The variation of T~(Ja/J) can be fitted to the linear law: k T J J = 2.48 J2/J + 1.76 (linear regression coefficient, r --- 0.9997) (Fig.
too short to influence the short-range magnetic order. The variation re(J1//J) can be fitted to a linear law: kTc/J= 0.71 J~/J+ 3.55 (r = 0.9983) (Fig. 8). 3.2.3. Influence
of Js (J2 = Jl = J)
Increasing ']3 broadens the peak of the thermal capacity (0.6 ~< ,6 ~ 1.1) but the broadening is weaker than if J2 increases (Fig. 10(a)). The tail of the order parameter at high temperatures seems to be more
8).
X- J3 = O, • - J3 = J/4, O- J3 = 3J/4, O- J3 = 3J/2
(a) 0,8
0,7
0
X
0,6
0
0 0 0
0,5
×
0.4 0
O0
O®O o
X X
X XXX OOO
0.3 X
XO • X
•
XX.X
0
• oOoo
0 0 oOC~ O
X
O•
0 0
0
0
0
0 0 O O
0 O0
0.2
O
0,1 0 2
I
I
t
I
I
I
I
I
2.5
3
3.5
4
4,5
5
5,5
6
kT/J X- J3 = 0, • - J 3 = J/4, O- J3 = 3J/4, o - J3 = 3 J / 2
(b) I
%.%%° o
0.9 0.8 0.7 0.6 0.5 0.4
×
o o o % 0 0 o OOo% •
O O
•
×
o~.
•
X
0.3 0.2 0.1 o
o
O
I,I i ×
•
0 0
O
oo~ ",~oo Ooo~
2
3
4
5
I
6
~,
o
I
7
o
I
8
kT/J
Fig. 10. Temperaturedependenceof (a) the thermalcapacityand (b) the orderparameterfor severalvaluesof J3 (finiteoctagonaltilingof 4361 spins).
D. Ledue, J. Teillet/ JournaI of Non-CrystaIIineSolids 191 (1995)216-226 0,12
transition, the maximum in the thermal capacity is related to the ordering in the ferromagnetic clusters.
0.11
o .Jl+0
4. Conclusion
A
- j3.~0
&
0.1
A
A
0,09
A
&
>
A Zk
~D v
0.08
o
0.07 c, % ,2, ,2,
0.06
0.05
I 0,5
1
~
r
1,5
2
kT/J
Fig. t 1. Temperature dependence of the thermal capacity if J1 va 0 or ,/3 4:0 (finite octagonal tiling of 4361 spins).
pronounced upon increasing J3 (Fig. 10(b)). The variation of Tc(J3//J) c a n be fitted to a linear law: krc//J= 1.20 J3//J+ 3.04 ( r = 0.9991) (Fig. 8).
3.2.4. Cases J1 4:0 (J2 = "13 = O) and J~ 4=0 (J2 = Jr O) • The two cases Jx > 0 ( J 2 = "/3 = 0) and "/3 > 0 (J2 = J1 = 0) which correspond to ground states with only ferromagnetic clusters (see Section 1) provide thermal capacity curves showing a maximum. In agreement with the energy curves which are almost linear, the location of this maximum is difficult to determine (Fig. 11). The temperature, TI, corresponding to the point of maximum slope in magnetic energy has been estimated at kTi/J I = 0.95 4-0.05 and kTi/J 3 = 0.91 4- 0.05. Since such magnetic systems do not undergo a long-range-order magnetic =
225
Monte Carlo simulations are of great interest for spin systems on quasiperiodic tilings for which analytical calculations are difficult. These simulations reveal analogies between the Ising model in 2D periodic lattices and the Ising model in the quasiperiodic octagonal tiling: generally, k T J ( z ) J increases if ( z ) increases; kTc/J varies linearly versus each exchange integral Ji (i = 1, 2, 3) if Ji/J <~3 / 2 ( J is the value of the two exchange integrals which are constant); increasing J2 or "/3 broadens the thermal capacity peak. On the other hand, some differences due to the quasiperiodicity have been shown: kTc/(z)J is generally slightly higher in quasiperiodic tilings showing that the tendency to ferromagnetic ordering is higher in quasiperiodic lattices than in periodic ones; increasing J1 does not broaden the thermal capacity peak. Our data indicate that quasiperiodicity is favourable to ferromagnetic ordering. However, it is difficult to compare these numerical data with previous experimental results on quasicrystals which have been claimed to be ferromagnetic because the origin of magnetic properties of such compounds does not seem to be clearly understood. Indeed, ferromagnetism could come from facts which are irrelevant to the quasiperiodicity, such as non-quasiperiodic parasitic phases. In the near future, we plan to calculate the static critical exponents of the transition for some frustrated Ising spin systems on the 2D octagonal tiling.
References [1] R. Bellissent, F. Hippert, P. Monod and F. Vigneron, Phys. Rev. B36 (1987) 5540. [2] C. Berger and J.J. Prejean, Phys. Rev. Lett. 64 (1990) 1769. [3] L. Kandel and F. Hippert, J. Magn. Magn. Mater. 104-107 (1992) 2033. [4] R.A. Dunlap and V. Srinivas, Phys. Rev. B40 (1989) 704.
226
D. Ledue, J. TeiIIet/ Joumal of Non-Crystalline Solids 191 (i995) 216-226
[5] A.P. Tsai, A. Inoue, T. Masumoto and N. Kataoka, Jpn. J. Appl. Phys. 27 (1988) L2252. [6] D.P. Landau, Phys. Rev. B13 (1976) 2997. [7] D.P. Landau, J. Appl. Phys. 42 (1971) 1284. [8] C. Fan and F.Y. Wu, Phys. Rev. 179 (1969) 560. [9] J. Tobochnik and G.V. Chester, Phys. Rev. B20 (1979) 3761. [10] J.E. van Himbergen and S. Chakravarty, Phys. Rev. B23 (1981) 359. [11] B. Berge, H.T. Diep, A. Ghazali and P. Lallemand, Phys. Rev. B34 (1986) 3177. [12] D. Ledue, J. TeiUet, J. Carnet and J. Dujardin, J. Non-Cryst. Solids 153&154 (1993) 403. [I3] M. Duneau, F. Dunlop, A. Jagannathan and C. Oguey, Mod. Phys. Lett. B5 (1991) 1895. [14] E.S. Sorensen, M.V. Jaric and M. Ronchetti, Phys. Rev. B44 (1991) 9271. [i5] Y. Okabe and K. Niizeki, J. Phys. Soc. Jpn. 57 (1988) 16. [16] Y. Okabe and K. Niizeki, J. Phys. Soc. Jpn. 57 (1988) 1536. [17] N.W. Dalton and D.W. Wood, J. Math. Phys. 10 (1969) 1271. [18] R.W. Gibberd, J. Math. Phys. 10 (1969) 1026. [19] D. Gratias, Du Cfistal 5 l'Amorphe (Editions de Physique, Paris, 1988).
[20] D. Ledue, 'Simulation de structures magn,~tiques dans des r6seaux quasip~riodiques bidimensionnels', thesis, University of Rouen (1991). [21] S. Kirkpatrick, C.D. Gelatt Jr and M.P. Vecchi, Science 220 (1983) 671. [22] N. Metropolis, A.E. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087, [23] M.E. Fisher and A.E. Ferdinand, Phys. Roy, Lett. 19 (1967) 169. [24] A.E. Ferdinand and M.E. Fisher, Phys. Roy, 185 (1969) 832. [25] M.E. Fisher, in: Prec. Int. Summer School Enrico Fermi 1970, Course 51, Varenna, Italy (Academic Press, New York, 1971); M.E. Fisher and M.N. Barber, Phys. Roy. Lett. 28 (1972) 1516. [26] H. Au-Yang and M.E. Fisher, Phys. Rev. Bll (1975) 3469. [27] K. Binder, Physica 62 (1972) 508. [28] D. Ledue, D.P. Landau and J. Teillet, Phys. Rev. B51 (1995) 12523. [29] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1972) 1181. [30] A. Herpin, TMorie du Magn~tisme (Presses Universitaires de France, Paris, 1968).
NON¢ gT,T,T S0IN ELSEVIER
Journal of Non-CrystallineSolids 191 (1995) 216-226
Critical temperature of the non-frustrated ferromagnetic Ising model on the quasiperiodic octagonal tiling D. Ledue *, J. Teillet LMA URA CNRS808, Facult~des Sciences, Universit~de Rouen, 76821 Mont-Saint-Aignanc~dex, France
Received29 June 1994;revisedmanuscriptreceived22 March 1995
Abstract The critical temperature of some non-frustrated ferromagnetic Ising spin systems on the two-dimensionaloctagonal tiling is calculated by Monte Carlo simulations using the simulated annealing method. The ferromagnetic interactions are limited to the third neighbours (J1, ./2, J3) where only J2 is a percolating interaction. The infinite-tiling critical temperature is estimated at kTc/J = 2.39 + 0.02 (J2 = J and Jl = J3 = 0), kTe/J = 3.05 --I-0.03 (J2 = Jl ~=J and J3 = 0), kTe/J = 3.60 + 0.04 (J2 = J3 = J and J1 = 0) and kTJJ = 4.39 + 0.04 (./2 = J1 = J3 = J). The value of kTo/(z) J is generally slightly higher in the octagonal tiling than in the square and triangular lattices indicating that the tendency to ferromagnetic ordering is higher in quasiperiodic tilings. It is found that the critical temperature, Tc, varies linearly with the different exchange integrals.
1. Introduction Some experiments on the magnetism of the icosahedral alloys have shown spin-glass-like properties at very low temperature for AI-Mn and AI-Mn-Si (see, for example, Refs. [1-3]). Later, weak ferromagnetic quasicrystals were claimed to be found [4,5]. In particular, A14oMn25F%CuvG%5 is ferromagnetic at 300 K and the Fe atoms have a weak moment (0.11/z B) [4]. Other investigations have shown ferromagnetic quasicrystals with higher Curie temperature (Tc = 533 K for A1525Ge22.sMn25) [5]. On the other hand, while classical spin systems on two-dimensional (2D) periodic lattices have been
* Correspondingauthor. Tel: +33 35 14 68 77. Telefax: +33 35 14 66 52. E-mail: denis.ledue@univ_rouen.fr.
widely studied (see, for example, Refs. [6-8] for the Ising model and Refs. [9-11] for the XY model), only a few studies have been devoted to 2D quasiperiodic systems [12-14]. For the XY model, numerical simulations on the Penrose tiling provided some frustrated magnetic ground states and indicated a Kosterlitz-Thouless-like transition [12]. For the frustrated Ising model on the octagonal tiling, analytical calculations have indicated some long-rangeordered magnetic ground states [13]. Also in the Penrose tiling, the magnetic transition of nonfrustrated ferromagnetic Ising spin systems has been studied using numerical simulations [14-16]. The results show that the Ising model on the Penrose tiling belongs to the same universality class as the Ising model on periodic lattices. Further, the critical temperature of the infinite Penrose tiling has been estimated as k T c / J = 2.401 + 0,005 [14] which is
0022-3093/95/$09.50 © 1995 ElsevierScienceB.V. All rights reserved SSDI 0022-3093(95)00246-4
D. Ledue, Jr. Teillet/ Journal of Non-Crystalline Solids 191 (1995) 216-226
higher than the critical temperature of the infinite square lattice ( k T c / J = 2.269 [6]) with the same mean number of interacting neighbours (( z ) = 4). In this paper, we present calculations of the critical temperature of the Ising model on another infinite quasiperiodic tiling: the octagonal tiling. Our study is carried out by numerical simulations and the data are compared with analytical results previously obtained on periodic lattices [17,18]. The octagonal tiling exhibits six different types of local environment (Fig. l(a)). The ferromagnetic ex-
217
t
a) Fig. 2. Some clusters due to JI or J3 (0, spins which belong to a Jl-cluster of four spins; [3, spins which belong to a Jl-cluster of eight spins; 0, spins which belongto a J3-clusterof 24 spins; O, spins which belong to a J3-clusterof 72 spins).
A
B
C
D
E
F
Fig. 1. (a) The octagonal tiling and the six local environments. (b) The exchange interactions (J2 correspondsto the edges of the unit cells).
change interactions have been limited to the third neighbours (J1/> 0, ,/2 >~ 0, J3 >~ 0) (Fig. l(b)). Unlike periodic lattices, all the exchange interactions do not allow magnetic percolation: only J2 favours alignment of all spins while J1 and J3 can only connect the spins into ferromagnetic clusters (Fig. 2). Then, we can deduce that the ground states are (i) ,/2 ~ 0: ferromagnetic ground state; (ii) ,/2 = 0: (a) J1 va 0 and "/3 4~ 0: ground states made up of two disconnected ferromagnetic sublattices and isolated spins (sites A) (Fig. 3); (b) J1 -- 0 and "/3 va 0: ground states made up of ferromagnetic clusters and isolated spins (sites A); (c) J1 4~ 0 and J3 = 0: ground states made up of small ferromagnetic clusters (maximal size: eight spins) and isolated spins (sites A, B, C, D). Different sets of positive exchange integrals on finite octagonal tilings have been used in order to study the influence of each exchange integral. The models, simulation technique and data analysis method are described in Section 2. The results and a discussion are presented in Section 3.
218
D. Ledue, J. TeiIlet / Journal of Non-Crystalline Solids 191 (1995) 216-226
2. Background 2.1. The octagonal tiling The quasiperiodic octagonal tiling exhibits an eight-fold non-crystallographic symmetry [19]. The 4D integral representation of C s, the eightfold cyclic group, decomposes into two inequivalent 2D representations: one operates on EEl, the physical space, the other on the orthogonal complementary space,
E.L. The 2D octagonal tiling is obtained by projecting the nodes of the original lattice, 774, which have a projection in E . falling within the octagonal window [13]. All the octagonal tilings belong to the same local isomorphism class, therefore the infinitetiling critical temperature does not depend on the tiling [20]. So, we have considered the octagonal tiling with a perfect eightfold symmetry around its centre. This tiling is generated using in E± an octagonal window which is centered on the origin. 2.2. Model and magnetic quantities
a)
Let each vertex of the octagonal tiling be occupied by an Ising spin, S t = + 1. The magnetic energy, U, and total magnetization, M, (in g/z B, where g is the Land6 factor and /xB is the Bohr magneton), are respectively given by
v=-
E s,is,sj, (i,j)
M = ~ S i. i The thermal capacity, C(T), is deduced from the fluctuations in the magnetic energy: c(r)
b)
(u 2 ) - ( U ) 2
ou
kT 2
OT'
=
where 1
(U) =
Z(T)
EV(c) exp(-V(c)/kT) ¢
is the thermal average (Z(T) = Ec e x p ( - U(c)/kT) is the partition function and the sum goes over all configurations, c). 2.3. Numerical simulation
Fig. 3. Ground states with Jl 4= 0, J3 @ 0 and '/2 = 0: (a) the two sublattices in opposite directions, (b) the two sublattices in the same direction (QIDO, sites A; e, sites B; ©, sites C; n, sites
D; 1,, sites E).
The procedure is a simulated annealing method [21]. In this Monte Carlo (MC) procedure, the magnetic energy is minimized by adjusting the direction of the spins in such a way as to freeze the system into a magnetic ground state. At each temperature, thermal equilibrium is achieved after a few thousands MC steps per spin following the Metropolis algorithm [22]. The spins are examined one after another and the energy variation, AU, which corre-
D. Ledue, J. Teillet / Journal of Non-CrystaIIine Solids 191 (1995) 216-226
219
(b) o.7
(8_) 0.7
i
0.6-
0.6 0.5 0.5
•
0,4 v
O o
o
0,4
• 0.3
o
••
t o
o
o
o
•4361
o
O o
o ,:3
0.2
O
e••
01
• 1433
0
O o
002481
• ••• t ••••
0,1
÷
f o
0.2
• 0.3
•4361
too
o
o
02929
'i'
::iii
O{
•
O O O O
•1169
o o o 0689 0
oo481
J
0 2
2,5
2.5
3
3.5
3
kT/J
kT/J
(d)
(O) 0.6
i.
0.5
0.6
0,5
o 0.4
0,4
°i
° o
0
o
o
•
•
•
o o
0.2
o
O
0,2
o
o
(3
O
o
o
-
O
o2929
o
o o
02929
0.1
• °1L• •
0,1
:: 0
0 2.5
3
3.5 kT/J
•4361
O
•4361
O
O
o
•
0
0 •
o
o
>~ 0,3
0.3
l°
4
3.5
o 0 o o
f
o o o
•
•1169
o o o 0689
;
J
I
4
4.5
5
kT/J
Fig. 4. Temperature variation of the thermal capacity for several lattice sizes ((a) (0, J, 0), (b) (J, J, 0), (c) (0, J, J), (d) (J, J, J)).
220
D. Ledue, J. TeiIIet/ Journal of Non-Crystalline Solids 191 (1995) 216-226
4.5
~
4 ° ~ 0
0
3.5 ?~"--~ " 3
~'(:~:X>
*""*'~
~
,
,
~o(J,J,J)
+ ' ' " ~
~ * ( O , J , J ) o(J,J,O)
+~°~°"
O"----.
2,5 2 1.5
-&(O,J,O) I
~
I
4
0.05
0,1
0.15
0,2
1,1"1/2 Fig. 5. Extrapolationof kTc(n)/J v e r s u s n -1/2.
sponds to the spin-flip is calculated. If the spin-flip decreases the magnetic energy of the system (AU ~< 0), it is accepted. If it increases the energy (AU > 0), it is not systematically rejected: the transition occurs with probability W ( A U ) = e x p ( - A g / k T ) . The low-temperature equilibrium is reached by simulated annealing: during the process, T is slowly decreased and, if this annealing is performed slowly enough, the spin configuration which is frozen will correspond to one of the ground states. Non-frustrated spin systems are ergodic, i.e., at thermal equilibrium, the average over MC steps converges towards the thermal average. For each run, 5000 MC steps per spin have been discarded before averaging U and M over the next 15 000 MC, steps per spin. Numerical simulations were carried out on finite octagonal tilings of different sizes (n = 334361) with free boundary conditions as in Ref. [12].
Table 1 Infinite-tilingcritical temperaturein the four cases of exchange interactions (mean numbers of interactingneighboursare analytical results) Jl .I2 J3 (z) kTc/ J 0 J 0 4 2.39,--0.02 J S 0 (4+ 6~-)/(1 + ~-) 3.05,--0.03 0 J J (8+ 4~')/(i+ ~ ) 3.60,-,0.04 J J J (8 + 6~')/(1 + V~") 4.39 _ 0.04
The cooling rate, ~-, was choosen equal to 0.98 so that simulations provide regular thermal capacity curves for each size. Numerical simulations were performed on a DG MV 15000-20 computer. 2.4. Calculation method of the critical temperature To determine the infinite-tiling critical temperature, we have used the 'finite-tiling transition temperature', To(n), which corresponds to the maximum in the thermal capacity or, equivalently, the point of maximum slope in magnetic energy [23-27]. The infinite-tiling critical temperature can be calculated by extrapolating the asymptotical linear variation of Table 2 Comparisonof kTc / ( z) J in quasiperiodictilingswith kTc / ( z ) J in periodiclattices Lattice (z) kTe / (z) J Square 4 0£67 [6,17,18] 8 0.657 [17] Triangular 6 0,608 [17] 12 0.731 [17] Penrose 4 0.600::1:0.001 [14] Octagonal 4 0.5974-0.006 (4+ 6V~-)/(I+ ~ ) 0,5904-0.006 (8+4V~')/(I+ ~ ) 0.636:I:0.006 (8+ 6~')/(I + f2") 0.643+-0,006
D. Ledue, J. TeilIet/ JournaI of Non-Crystalline Solids 191 (1995) 216-226
QP lattices
o
P lattices
0.9
•
Beth• theory N
221
- ' " ""
,,.,''* oo
.o•
0.8
....--
..-'''""
v
0.7
..'" o ._o2, ~
0.6
A/ l
I
I
l
I
I
2
4
6
8
10
12
0.5 0
Fig, 6, Comparisonof kTc/(z) J in quasiperiodic filings with kTc/{ z)J in periodic lattices. O- J2= J / 4 , 0 - J 2 = 3 J / 4 , O - J 2 = J , o - J 2 = 3 J / 2
(a) 0.7 0.6
O0
@O
0.5 > o
O O
',
O
**• • 0.
•
0
O
•
O
O
0
0.4 #
0.3
0 O O0
•
0• ,00 •
0
• 000
~o~O0 ° •*
00 0 0
%
0.2 0.1 0
,
1
I
E
I
I
l
I
2
3
4
5
6
7
kT/J • - J2 = J / 4 , <>- J2 = 3 J / 4 , O- J2 = J, O- J 2 = 3 J / 2
(b) 1
0
•
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 o.1
*
0
OOo ".... °°Ooo
\ •
0
0 •
0
•
0
0
0 <;,
•
0
}..
0
o 1
1
I
2
3
• •
<>
•
°
•
00
0 •
4
5
o° ooo 6
•
o
o
o
i
i
7
8
kT/J
Fig. 7. Temperature dependence of (a) the thermal capacity and (b) the order parameter for several values of J2 (finite octagonal tiling of 4361 spins).
222
D. Ledue, J. TeilIet/ Journal of Non-CrystallineSolids 191 (199512•6-226
To(n) versus n -llz~ (v is the critical exponent associated to the correlation length) [6]. Numerical simulations on the ferromagnetic Ising model on the octagonal tiling provided v = 1 which is the 2D periodic value [28]. For a given size and a given set of exchange integrals, it has been shown that several numerical simulations provide similar thermal capacity curves. So, the accuracy on the infinite-tiling critical temperature is not limited by the statistical fluctuations from one numerical simulation to another but by the accuracy in the estimate of the location r~(n) of the maximum in the thermal capacity. Then, the uncertainty has been estimated at ATo = ½(1 - 'r)T~, where ¢ is the cooling rate.
3. Results
tiling is generally higher than in the square and triangular ones (Fig. 6). This difference can be explained assuming that quasiperiodicity is less fair to the propagation of the spatial fluctuations of the local magnetic field than periodicity. It should be noted that, according to the accuracy of our data, we cannot conclude if adding the nearest neighbour interaction, J1, changes the value of k T d ( z ) J (Fig. 6 and Table 2). This look of change can be explained by the fact that J1 has a small influence on the transition temperature because it can only connect the spins into small ferromagnetic clusters (maximal size: eight spins). On the other hand, taking into account the interaction with the third neighbours, J3, increases k T c / ( z ) J although J3 (as Jl) can only connect the spins into ferromagnetic clusters (nonpercolating interactions). This effect is attributed to the mean size of the clusters due to J3 (J3 can
As expected, since the transition should be ferromagnetic (J2 @ 0), C / k ~ 0 as T ~ OK ( C / n k 1/2 for the Kosterlitz-Thouless transition [12,29]). The size dependence of the order parameter data ((IMI)n) is quite pronounced even well below T~(n) as in the square lattice with free edges [6].
• - vs J2/J ~
/ .,,.,,X
,Di~-"~ 7 x
The infinite octagonal tiling critical temperature has been calculated in four cases which correspond to four sets of exchange interactions (J1, J2, -/3): The temperature variation of the thermal capacity for several lattice sizes indicates that the maximum in the thermal capacity, Cm,~×, increases with the lattice size (Fig. 4). The asymptotical behaviour of To(n) versus n -I/2 ( n - I / 2 ~ 0) is shown in Fig. 5. The infinite-tiling critical temperature for each case is given in Table 1. If ( z ) = 4, the octagonal tiling and the Penrose tiling exhibit close critical temperatures [14]. However, since the octagonal tiling and the Penrose tiling are not locally isomorphic, we cannot conclude that these temperatures are equal. In order to study the influence of the quasiperiodicity, the values of k T c / ( z ) J for several periodic and quasiperiodic lattices are regrouped in Table 2. Our simulations show that k T o / ( z ) J in the octagonal
/
x . vs J3/J
3.1. Infinite-tiling critical temperature - - comparison between periodic and quasiperiodic lattices
(o, J, 0), (J, J, o), (o, s, J), (J, s, s).
/ *
-VSJl/J
4 . -'Z''"
×1 i
,/
f 3
1//~l'/
×
0
/
/
l,
I-
I
I
1
1
I
0.25
0,5
0,75
1
1,25
1.5
Fig. 8. Variationof kTe/J versus the exchangeintegrals (finite octagonaltiling of 4361 spins).
D. Ledue, J. TeilIet/ Journal of NoniCrystalline Solids 191 (1995)216-226
223
3.2. Influence of each exchange integrals (n = 4361)
connect the spins into large clusters which surround the centre of the tiling) (Fig. 2). As expected, our values of k T c / ( z ) J are lower than those given by the Bethe theory (kTc/zJ = 2 / ( z In(z/z - 2))) [30], which partially neglects the local magnetic field fluctuations near the transition (Fig. 6). It should be noted that in the square lattice with ( z ) = 8, a lower value k T J ( z ) J = 0.620 has been found by Gibberd [18]. Unlike Ref. [17] in which the authors have used series expansion, Gibberd has considered the freefermion approximation which has been used previously in the 2D ferroelectric problem.
In this section, calculations are restricted to a decent-sized finite octagonal tiling ( n - - 4 3 6 1 ~ [T~ - T~(4361)]/T~ - 0.02) (Fig. 5). The transition temperature has been determined in three cases in order to study the influence of each exchange integral: (i) influence of ,/2: J 1 = J3 = J and J / 4 <~J 2
3J/2 (the case '/2 = 0 which is slightly different has been excluded (see Section 1)); (ii) influence of Jl: 72 = J3 = J and 0 < JI ~< 3J/2;
X - J1 = O, • - J1 = J / 4 , O - J1 = 3 J / 4 , 0 - J1 = 3 J / 2
(a) 0.7 0.6
•
× X
0,5
X
O
O< • X
0.3
x
XXo
0 o
XO
•
0
0
0 0
0 0
X
<> 0 •
0
0
o,
•
X
0
0
•
,,
0,4
o
0
0 0
oOO
X
• X
¢
•
0
0
0.2 0.1 0
3
I
I
3.5
4
l
I
I
4,5
5
5.5
kT/J x- J l = 0 , O - J l = J / 4 , O -
Jl=3J/4, o- Jl=3J/2
(b) I X •
0.9
X@
~
0.8
OOt~0 0000,~^ •~OO 0 ~uO 0
Xx~',o % ^
0.7
"'Xx • X
0.6
X
Oo
~O •
UO O
0
•
0
0
X
i o.5
,5
0.4 0.3 0.2 0.1 0
I O
•
0
X @
0
Xx~l~,• • I P
2
3
4
,"
@<> X i
5
00¢><
0
6
O E
7
kT/J
Fig. 9. Temperature dependence of (a) the thermal capacity and (b) the order parameter for several values of Jl (finite octagonal tiling of 4361 spins).
D. Ledue,J. Teillet/ JournaIof Non-CrystallineSolids191 (1995)216-226
224
(iii) influence of J 3 : J 2
Influence of JI (J2 = "Is = J) Increasing J1 does not broaden the peak of the thermal capacity (A = 0.9) and does not change the tail of the order parameter above T~ (Fig. 9) because the interaction length of the NN interaction, Jl, is
3,2.2.
= J1 = J and 0 ~
3:/2. 3.2.1. Influence o f J 2 (J~ = J3 = J) The temperature dependence of the thermal capacity and the order parameter for several values of J2 are drawn in Fig. 7. Increasing J2 broadens the peak of the thermal capacity (0.5 ~< A ~ 1.3; A is the width at half-height) and is responsible for a more pronounced tail of the order parameter at high temperatures. This tail is attributed to the short-range magnetic order above T~. The variation of T~(Ja/J) can be fitted to the linear law: k T J J = 2.48 J2/J + 1.76 (linear regression coefficient, r --- 0.9997) (Fig.
too short to influence the short-range magnetic order. The variation re(J1//J) can be fitted to a linear law: kTc/J= 0.71 J~/J+ 3.55 (r = 0.9983) (Fig. 8). 3.2.3. Influence
of Js (J2 = Jl = J)
Increasing ']3 broadens the peak of the thermal capacity (0.6 ~< ,6 ~ 1.1) but the broadening is weaker than if J2 increases (Fig. 10(a)). The tail of the order parameter at high temperatures seems to be more
8).
X- J3 = O, • - J3 = J/4, O- J3 = 3J/4, O- J3 = 3J/2
(a) 0,8
0,7
0
X
0,6
0
0 0 0
0,5
×
0.4 0
O0
O®O o
X X
X XXX OOO
0.3 X
XO • X
•
XX.X
0
• oOoo
0 0 oOC~ O
X
O•
0 0
0
0
0
0 0 O O
0 O0
0.2
O
0,1 0 2
I
I
t
I
I
I
I
I
2.5
3
3.5
4
4,5
5
5,5
6
kT/J X- J3 = 0, • - J 3 = J/4, O- J3 = 3J/4, o - J3 = 3 J / 2
(b) I
%.%%° o
0.9 0.8 0.7 0.6 0.5 0.4
×
o o o % 0 0 o OOo% •
O O
•
×
o~.
•
X
0.3 0.2 0.1 o
o
O
I,I i ×
•
0 0
O
oo~ ",~oo Ooo~
2
3
4
5
I
6
~,
o
I
7
o
I
8
kT/J
Fig. 10. Temperaturedependenceof (a) the thermalcapacityand (b) the orderparameterfor severalvaluesof J3 (finiteoctagonaltilingof 4361 spins).
D. Ledue, J. Teillet/ JournaI of Non-CrystaIIineSolids 191 (1995)216-226 0,12
transition, the maximum in the thermal capacity is related to the ordering in the ferromagnetic clusters.
0.11
o .Jl+0
4. Conclusion
A
- j3.~0
&
0.1
A
A
0,09
A
&
>
A Zk
~D v
0.08
o
0.07 c, % ,2, ,2,
0.06
0.05
I 0,5
1
~
r
1,5
2
kT/J
Fig. t 1. Temperature dependence of the thermal capacity if J1 va 0 or ,/3 4:0 (finite octagonal tiling of 4361 spins).
pronounced upon increasing J3 (Fig. 10(b)). The variation of Tc(J3//J) c a n be fitted to a linear law: krc//J= 1.20 J3//J+ 3.04 ( r = 0.9991) (Fig. 8).
3.2.4. Cases J1 4:0 (J2 = "13 = O) and J~ 4=0 (J2 = Jr O) • The two cases Jx > 0 ( J 2 = "/3 = 0) and "/3 > 0 (J2 = J1 = 0) which correspond to ground states with only ferromagnetic clusters (see Section 1) provide thermal capacity curves showing a maximum. In agreement with the energy curves which are almost linear, the location of this maximum is difficult to determine (Fig. 11). The temperature, TI, corresponding to the point of maximum slope in magnetic energy has been estimated at kTi/J I = 0.95 4-0.05 and kTi/J 3 = 0.91 4- 0.05. Since such magnetic systems do not undergo a long-range-order magnetic =
225
Monte Carlo simulations are of great interest for spin systems on quasiperiodic tilings for which analytical calculations are difficult. These simulations reveal analogies between the Ising model in 2D periodic lattices and the Ising model in the quasiperiodic octagonal tiling: generally, k T J ( z ) J increases if ( z ) increases; kTc/J varies linearly versus each exchange integral Ji (i = 1, 2, 3) if Ji/J <~3 / 2 ( J is the value of the two exchange integrals which are constant); increasing J2 or "/3 broadens the thermal capacity peak. On the other hand, some differences due to the quasiperiodicity have been shown: kTc/(z)J is generally slightly higher in quasiperiodic tilings showing that the tendency to ferromagnetic ordering is higher in quasiperiodic lattices than in periodic ones; increasing J1 does not broaden the thermal capacity peak. Our data indicate that quasiperiodicity is favourable to ferromagnetic ordering. However, it is difficult to compare these numerical data with previous experimental results on quasicrystals which have been claimed to be ferromagnetic because the origin of magnetic properties of such compounds does not seem to be clearly understood. Indeed, ferromagnetism could come from facts which are irrelevant to the quasiperiodicity, such as non-quasiperiodic parasitic phases. In the near future, we plan to calculate the static critical exponents of the transition for some frustrated Ising spin systems on the 2D octagonal tiling.
References [1] R. Bellissent, F. Hippert, P. Monod and F. Vigneron, Phys. Rev. B36 (1987) 5540. [2] C. Berger and J.J. Prejean, Phys. Rev. Lett. 64 (1990) 1769. [3] L. Kandel and F. Hippert, J. Magn. Magn. Mater. 104-107 (1992) 2033. [4] R.A. Dunlap and V. Srinivas, Phys. Rev. B40 (1989) 704.
226
D. Ledue, J. TeiIIet/ Joumal of Non-Crystalline Solids 191 (i995) 216-226
[5] A.P. Tsai, A. Inoue, T. Masumoto and N. Kataoka, Jpn. J. Appl. Phys. 27 (1988) L2252. [6] D.P. Landau, Phys. Rev. B13 (1976) 2997. [7] D.P. Landau, J. Appl. Phys. 42 (1971) 1284. [8] C. Fan and F.Y. Wu, Phys. Rev. 179 (1969) 560. [9] J. Tobochnik and G.V. Chester, Phys. Rev. B20 (1979) 3761. [10] J.E. van Himbergen and S. Chakravarty, Phys. Rev. B23 (1981) 359. [11] B. Berge, H.T. Diep, A. Ghazali and P. Lallemand, Phys. Rev. B34 (1986) 3177. [12] D. Ledue, J. TeiUet, J. Carnet and J. Dujardin, J. Non-Cryst. Solids 153&154 (1993) 403. [I3] M. Duneau, F. Dunlop, A. Jagannathan and C. Oguey, Mod. Phys. Lett. B5 (1991) 1895. [14] E.S. Sorensen, M.V. Jaric and M. Ronchetti, Phys. Rev. B44 (1991) 9271. [i5] Y. Okabe and K. Niizeki, J. Phys. Soc. Jpn. 57 (1988) 16. [16] Y. Okabe and K. Niizeki, J. Phys. Soc. Jpn. 57 (1988) 1536. [17] N.W. Dalton and D.W. Wood, J. Math. Phys. 10 (1969) 1271. [18] R.W. Gibberd, J. Math. Phys. 10 (1969) 1026. [19] D. Gratias, Du Cfistal 5 l'Amorphe (Editions de Physique, Paris, 1988).
[20] D. Ledue, 'Simulation de structures magn,~tiques dans des r6seaux quasip~riodiques bidimensionnels', thesis, University of Rouen (1991). [21] S. Kirkpatrick, C.D. Gelatt Jr and M.P. Vecchi, Science 220 (1983) 671. [22] N. Metropolis, A.E. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087, [23] M.E. Fisher and A.E. Ferdinand, Phys. Roy, Lett. 19 (1967) 169. [24] A.E. Ferdinand and M.E. Fisher, Phys. Roy, 185 (1969) 832. [25] M.E. Fisher, in: Prec. Int. Summer School Enrico Fermi 1970, Course 51, Varenna, Italy (Academic Press, New York, 1971); M.E. Fisher and M.N. Barber, Phys. Roy. Lett. 28 (1972) 1516. [26] H. Au-Yang and M.E. Fisher, Phys. Rev. Bll (1975) 3469. [27] K. Binder, Physica 62 (1972) 508. [28] D. Ledue, D.P. Landau and J. Teillet, Phys. Rev. B51 (1995) 12523. [29] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1972) 1181. [30] A. Herpin, TMorie du Magn~tisme (Presses Universitaires de France, Paris, 1968).