Table Of ContentIsing Dipoles on the Triangular Lattice
U.K. Ro¨ßler
IFW Dresden, Institut fu¨r Metallische Werkstoffe,
Helmholtzstraße 20, D-01069 Dresden, Germany.
(October 10, 2000)
1
0
0
2 perpendicularmomentsoftheparticles. Allthecouplings
AclusterMonteCarlomethodforsystemsofclassicalspins
n are antiferromagnetic. Length is measured in nearest-
with purely dipolar couplings is presented. It is tested and
a neighbour distances. If we restrict the interactions in
J applied for finitearrays of perpendicular Isingdipoles on the (1) to nearest-neighbourterms we get anIsing antiferro-
triangular lattice. This model is a modification with long-
6 magnet. The triangular Ising antiferromagnet (TIAFM)
2 range interactions of the geometrically frustrated Ising anti- is one of the few exactly solved systems with geometri-
ferromagnet. From measurements of integrated autocorrela- cal frustration5. It was shown that no long-range order
] tion times for energy, magnetization and staggered magne-
h is established at finite temperature, but the system be-
tizations, a high efficiency of the cluster Monte Carlo (MC)
c comes critical at T = 0. However, additional interac-
algorithm compared to a single-spin-flip algorithm is found.
e tions may lift the ground-state degeneracy and may in-
m Fortheinvestigated model, a finitetemperature transition is
ducephase-transitionsinsuchunderconstrainedsystems.
found which is characterized by a peak in the specific heat
at- and in thestaggered susceptibilities. HeCnlcues,tmero-MdeCl (m1)etishoindtserfeosrtisnpgininmiotsdeolws6n–8riagrhet.only effi-
st 75.40.Mg,75.50.Tt,02.70.Lq cient in critical regions if the clusters do not percolate
. already above the critical temperature. The energy of
t
a model (1) may be decomposed into two parts
m The progress in the fabrication of very small patterns
d- osifbfielirtryomtoagsntuedtiyc mexapteerriimalesn1–ta4lloypesntasttishteicfaalscpirnoapteinrtgiepsoos-f H =XV∆+ 21 X′v(Rij), (2)
n ∆ (i,j)6∈NN
some mesoscopic or exotic magnetic systems. Dipolar
o
c couplings between such particles may induce magnetic where the sum of V∆ denotes sums of nearest-neighbour
[ ordering3,4. These couplings are often only viewed as interactionsin one ofthe two types oftriangularplaque-
additional effective anisotropy barriers for the super- ttes (and, with open boundaryconditions, somenearest-
1
paramagnetic relaxation of individual particles. How- neighbour interactions along the boundaries also labeled
v
2 ever,such assemblies of ferromagneticnano-particlesare byV∆). ThisfirstpartisexactlytheTIAFM.Thesecond
1 coupled magnetic systems. They should display mag- sum contains the remaining bonds with longer range.
4 netic phase-transitions for systems with mesoscopic size Clusters for the TIAFM can be defined by sampling
1 of individual magnetic moments. Recently, Wirth et the sums of energies corresponding to elementary trian-
0 al. studied the properties of nanoscale ferromagnetic gular plaquettes9. These clusters are the correct critical
1
columns which were grown on substrates in a scan- clusters of the TIAFM percolating at T = 0. Applying
0
ning tunnel-microscope3. Magnetic-force microscopic this cluster definition to the first part of (2) the dipolar
/
t pictures of densely packed columns on a triangular lat- system is mapped on a system of clusters which interact
a
m tice showed typical antiferromagnetic correlations where throughthelong-rangeinteractionsgeneratedfromterms
rows of nearest-neighbour moments point alternatingly in the second sum of (2). For flipping clusters the usual
-
d up and down. Metropolisalgorithmis used. The algorithmofalternat-
n Thispaperpresentsanumericalstudyontheexpected ingly mapping (1) onto interacting clusters via (2) and
o thermodynamics for such assemblies of perpendicular sampling this mapped system is equivalent to sampling
c dipoles in two dimensions. Its second aim is the intro- the original system.
:
v duction ofa cluster Monte Carlo(MC) method for dipo- Wehavetoshowergodicityanddetailedbalancetojus-
i larly coupled systems. For the geometrically frustrated tifythemethod. Theprocedureofclusterdecomposition
X
triangular lattice, this method allows efficient statisti- may result in deleting all bonds which maps the origi-
r
a cal sampling down to relatively low temperatures. The nal system onto itself. Thus, the occurrence of single-
coupled system we want to study consists of i = 1...N spinflipsensuringergodicitycanbecheckeda posteriori.
Ising–spins, σi 1,1 Detailed balance may be demonstrated by the prescrip-
∈{− }
tion for general cluster algorithms given by Kandel and
H = 1X′v(Rij)σiσj, (1) Domany8whichshowsthatbondsmaybedeleted,frozen,
2 orleftunchangedtogeneratesystemsofinteractingclus-
i,j
ters equivalent to the original system.
where v(R) = gR−3 are interactions between three– The method is applied for finite arrays of hexago-
dimensional dipoles describing the coupling between the nal shape with N = 61...2977 spins. The follow-
1
ing measure for antiferromagnetic correlations is used: coolingandheatingwerefirstrelaxedforlongertimesbe-
Spins are labeled σ for sites given by µr +νr with forestartofmeasurementstoensurethattheyarecloseto
µν 1 2
nearest-neighbour vectors r = ( √3/2,1/2). Three equilibrium. Samplingwasperformedforenergyperspin
1,2
types of staggered magnetizations m±ay be calculated by E, specific heat C =N(<E2> <E>2)/T2, the stag-
−
mI = 1/N σ ( 1)µ, mII = 1/N σ ( 1)ν, gered magnetizations, the corresponding susceptibilities
and mIII =P1/µNν µν −σ ( 1)µ+ν. ForPthµeν hµexνa−gonal χX = N(<(mX)2> <mX>2)/T and the fourth-order
Pµν µν − Binder cumulant UX−= 1 <(mX)4> /(3 <(mX)2>2)
arrays, the three types are equivalent. Thus we use the
−
average m =1/3(mI+mII+mIII). (X = I, II, III). Using the histogram method by Ferren-
a berg and Swendsen11 these various quantities were eval-
Here, our algorithm is first used in the Swendsen-
uated as functions of temperature in the critical region.
Wang (SW) form by deleting/freezing bonds for all pla-
Results are shown in Figs. 2 and 3. The diverging peak
quettes of a randomly chosen type. It turns out that
in the specific heat (Fig. 2(b)) indicates the presence
after mapping onto the system of interacting clusters
of a transition at or closely below T/g 0.18. Simi-
best performance was achieved by attempts to flip just
≃
larly, the peak for the average staggered susceptibility
one (randomly chosen or the largest) cluster. Integrated
χ shows that the transition (Fig. 3(a)) is related to an-
auto-correlation times for energy, magnetization, and a
for the staggeredmagnetizationsmI...III were calculated tiferromagnetic ordering. Near T/g 0.18, ma steeply
≃
increases reaching values 0.3 to 0.4 (not shown here).
as measure of efficiency by a self-consistent windowing
∼
method10. Independent isothermal runs were started However,theaverageBinder-cumulantUaforthevarious
systemsizesdoesnotprovideaclearindicationforacon-
from slowly cooled and relaxed configurations. For sizes
N = 61...469 between several 104 and 105 cluster-MC tinuous transition by a unique crossing of Ua(T)-curves
for the different system sizes at one critical temperature
sweeps (MCS, i.e., N attempted cluster moves), for the
two larger sizes only 103...4 MCS could be performed. (Fig.3(b)). Thetransitionmightbefirstorderinsteadas
indicated by the observed hysteresis when cycling tem-
As expected, the autocorrelations for the magnetization
perature. Then the apparent correlation of system sizes
decay rapidly at all temperatures. The autocorrelations
N < 1000 corresponds only to the size of typical nuclei
for energy are similar to those of the staggeredmagneti-
of the low-temperature phase. On the other hand, the
zations. Thus, we only show the average integrated au-
behaviour of the U (T)–curves which seem to merge be-
tocorrelationtime τ forthe staggeredmagnetizationsin a
a
low T/g = 0.18, is consistent with usual observations at
Fig.1(a)and compareitto runs withthe single-spin-flip
a Berezinskii-Kosterlitz-Thouless-transitionanda line of
Metropolis algorithm for the two smallest system sizes.
critical points at low temperature. Possibly, the peak in
Fig.1(b) showsthe averagesize <c> of flipped clusters.
the specific heat is only a precursor of this transition, or
The strong increase of <c> towards low temperature
two transitions are present.
means that all spins are increasingly correlated. This
Inconclusiona cluster-MC method is presentedwhich
transition becomes sharper with larger N. The size dis-
maybeusedfordipolarlycoupledspin-systems. Itworks
tribution of flipped clusters always consists of a rapidly
bygeneratinginteractingclustersfromsamplingofshort-
decreasing part starting with single-spin flips. For low
range interactions. It is shown to be efficient for the
temperatures and small system sizes clusters with size
specific case of Ising dipoles on the triangular lattice.
close to N are found also.
Thissuccessreliesontheabilitytodefineclustersforthis
Still, the corresponding τ does not grow dramati-
a
geometrically frustrated system which percolate only at
cally. For N > 469 we could measure τ only down
a
T =0. However, the method can be generalized to vari-
to T/g 0.19, where it becomes essentially indepen-
≃ ousothersystemswithlong-rangeinteractions,including
dent of N while <c> 2. The single-spin-flip algorithm
≃ systemswith continuousspins7. The investigationofthe
yields far largerτ whichina criticalregionshouldgrow
a
with system size as τ Nz/2 with z 2. This com- exact nature of the finite-temperature transition found
a
∝ ≃ with this MC-method remains a task for the future.
parison demonstrates the efficiency of this cluster-MC.
Acknowledgment. I thank S. Wirth and A. Mo¨bius for
However, the numerical cost of the SW-algorithm scales
as Nτ . Therefore, a Wolff-type algorithm7 was imple- discussion. This work was supported by DFG. After fin-
a
ishingworkonthispaperIbecameawarethatdenHertog
mentedforfurtherproductionrunsshownbelow. Onlya
et al. have developed a similar cluster MC-method for
single cluster is grown starting from a randomly chosen
dipolar spin ice on the pyrochlore lattice12.
plaquette. The performance of this algorithm is similar
to the SW-algorithm but the numerical effort is smaller.
Its numerical cost scales as <c> τ .
a
Cluster-MCrunswithseveral105MCSwereperformed
for N 721. For N 1519 serious problems in the
≤ ≥
critical region T/g 0.18 arise. Cycling temperature
≤
displays hysteresis of the energy. Measurements could
be done for runs with only few 103 MCS in the critical 1A.D. Kent, T.M. Shaw, S. von Moln´ar, and D.D.
Awschalom, Science 262, 1249 (1993).
region with N 1519. Here, configurations both from
≥ 2M.R.Scheinfein,K.E.Schmidt,K.R.Heim,andG.G.Hem-
2
bree, Phys.Rev. Lett.76, 1541 (1996). -0.875
3S. Wirth, M. Field, D.D. Awschalom, and S. von Moln´ar,
Phys.Rev.B57,R14028(1998);S.Wirth,S.vonMoln´ar, (a)
J. Appl. Phys.87, 7010 (2000). -0.885
4R.P. Cowburn, A.O. Adeyeye, and M.E. Welland, New J.
Phys.1, 16 (1999). N=61
5G.H. Wannier, Phys.Rev. 70, 357 (1950). 217
6R.H. Swendsen and J.S. Wang, Phys. Rev. Lett. 58, 86 E/g -0.895 476291
(1987). 1519
7U.Wolff, Phys. Rev.Lett. 62, 1087 (1989). 2977
8D.Kandel and E. Domany,Phys.Rev.B 43, 8539 (1991). -0.905
9P.D. Coddington, L. Han, Phys. Rev. B 50, 3058 (1994);
G.M. Zhang, C.Z. Yang, Phys. Rev.B 50, 12 546 (1994).
10U.Wolff, Phys. Lett. B 228, 379 (1989). -0.915
11A.M.FerrenbergandR.H.Swendsen,Phys.Rev.Lett.63, 2
1195 (1989). (b)
12B.C. den Hertog, R.G. Melko, M.J.P. Gingras, cond- 1.75
mat/0009225 (unpublished).
1.5 N=2977
1519
eat 1.25 721
H 469
10000 MMeetrtoroppoolisli:s :N N==26117 cific 1 26117
e
p
S 0.75
1000
0.5
S] Cluster: N=2977
C N=1519
M 100 N=469 0.25
[a N=217
τ N=61 0
10 0.14 0.16 0.18 0.2
Temperature T/g
(a)
1 FIG.2. (a) Energy per spin (b) specific heat in the region
of the transition temperature of Ising dipoles on hexagonal
476291 N=26117 triangular–lattice arrays with various sizes N.
469
217 721
1519
100
61
>
c
<
10
(b)
1
0.1 1
Temperature T/g
FIG. 1. (a) Average integrated autocorrelation time for
staggeredmagnetizations(seetext)versustemperature. Sym-
bolscorrespondtoindependentisothermalruns. Comparison
betweenMetropolissingle-spin-flipandclusteralgorithm. For
the largest size N = 2977, only one run at T/g = 0.190 is
shown (big circle). (b) Corresponding average cluster sizes
<c>for different system sizes.
3
100
N=2977
(a)
1519
721
469
217
61
a 10
χ
1
0.6
N=61
217
469
721
Ua 1519
0.4 2977
(b)
0.2
0.14 0.16 0.18 0.2 0.22 0.24
Temperature T/g
I
FIG.3. (a) Average staggered susceptibility χa = 1/3 (χ
II II
+χ +χ ). (b) Corresponding average fourth-order Binder
cumulant in thecritical temperature region.
4
